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+{
+    "url": "http://arxiv.org/abs/2405.02228v1",
+    "title": "REASONS: A benchmark for REtrieval and Automated citationS Of scieNtific Sentences using Public and Proprietary LLMs",
+    "abstract": "Automatic citation generation for sentences in a document or report is\nparamount for intelligence analysts, cybersecurity, news agencies, and\neducation personnel. In this research, we investigate whether large language\nmodels (LLMs) are capable of generating references based on two forms of\nsentence queries: (a) Direct Queries, LLMs are asked to provide author names of\nthe given research article, and (b) Indirect Queries, LLMs are asked to provide\nthe title of a mentioned article when given a sentence from a different\narticle. To demonstrate where LLM stands in this task, we introduce a large\ndataset called REASONS comprising abstracts of the 12 most popular domains of\nscientific research on arXiv. From around 20K research articles, we make the\nfollowing deductions on public and proprietary LLMs: (a) State-of-the-art,\noften called anthropomorphic GPT-4 and GPT-3.5, suffers from high pass\npercentage (PP) to minimize the hallucination rate (HR). When tested with\nPerplexity.ai (7B), they unexpectedly made more errors; (b) Augmenting relevant\nmetadata lowered the PP and gave the lowest HR; (c) Advance retrieval-augmented\ngeneration (RAG) using Mistral demonstrates consistent and robust citation\nsupport on indirect queries and matched performance to GPT-3.5 and GPT-4. The\nHR across all domains and models decreased by an average of 41.93% and the PP\nwas reduced to 0% in most cases. In terms of generation quality, the average F1\nScore and BLEU were 68.09% and 57.51%, respectively; (d) Testing with\nadversarial samples showed that LLMs, including the Advance RAG Mistral,\nstruggle to understand context, but the extent of this issue was small in\nMistral and GPT-4-Preview. Our study con tributes valuable insights into the\nreliability of RAG for automated citation generation tasks.",
+    "authors": "Deepa Tilwani, Yash Saxena, Ali Mohammadi, Edward Raff, Amit Sheth, Srinivasan Parthasarathy, Manas Gaur",
+    "published": "2024-05-03",
+    "updated": "2024-05-03",
+    "primary_cat": "cs.CL",
+    "cats": [
+        "cs.CL",
+        "cs.AI",
+        "cs.IR"
+    ],
+    "label": "Original Paper",
+    "paper_cat": "LLM Fairness",
+    "gt": "REASONS: A benchmark for REtrieval and Automated citationS Of scieNtific Sentences using Public and Proprietary LLMs",
+    "main_content": "Introduction The development of LLMs marks a significant advancement in computational linguistics and artificial intelligence (AI) (Tamkin and Ganguli, 2021). LLMs, such as OpenAI\u2019s GPT series, have shown remarkable capabilities in text generation (Zhao et al., 2023), and question-answering systems (Rasool et al., 2023; Elgedawy et al., 2024). However, their limitations become apparent as they become more integrated into various domains, including defense (Schwinn et al., 2023), news media (Fang et al., 2023), and education (Yan et al., 2024; Hung et al., 2023; Augenstein et al., 2023). The critical issue is their propensity to generate hallucinated sentences and propagate factually inaccurate pieces of information without reference (Ji et al., 2023; Rawte et al., 2023). These inaccuracies diminish the models\u2019 reliability and erode users\u2019 trust, a vital component in their widespread adoption. Commercial LLM-based search systems, including Bing Search-powered GPT 4 (Mehdi, 2024) and Perplexity.ai (Roose, 2024), are still not capable enough of resolving the issue of citation generation to confirm the scientific feasibility of either a generated sentence(s) or given sentence(s) from the scientific literature. For instance, Figure 1 shows how proprietary LLMs respond to the zero-shot indirect query. It is evident from the figure that while general-purpose LLMs like GPT3.5 and GPT-4 \u2018pass\u2019 the query, task-specific LLM Perplexity does generate relevant citations but still shows hallucination. Consider the following arXiv:2405.02228v1  [cs.CL]  3 May 2024 \fFigure 1: An illustration and motivating example for investigating LLMs for automatic citation generation task. Perplexity.ai, which is an LLM-based search engine, yields a citation that doesn\u2019t exist [1], an incorrect one [3], and a correct citation [2]. Advance RAG (defined in this research) improved context understanding and citation generation quality. Time: Feb. 05, 2024. three real world examples of this research: Citation Generation in Research Articles and News Reports: LLMs can generate highly persuasive and realistic content, especially in writing research articles or news reports, making it challenging for users to distinguish between genuine and fabricated information Nakano et al. (2021); Menick et al. (2022); Kumarage and Liu (2023). Citation Generation in Reports for Organizational Cybersecurity: In cybersecurity, where decisions often need to be made quickly and are based on the data provided, the accuracy and reliability of information are paramount (Divakaran and Peddinti, 2024). Inaccurate citations can lead to misinformation and potentially severe consequences in decision-making processes. LLMs can automate the citation generation process but need to be carefully designed for organization specific cybersecurity. Citation Generation in Reports for Legal: In a significant event, an attorney tried employing ChatGPT for legal analysis during a trial (see subsection A.1)(Bohannon, 2023). While ChatGPT generated information, it failed to capture the nuanced complexities and critical legal precedents needed for the case. This underscores the importance of confirming and sourcing accurate legal citations and precedents relevant to the case. We contribute by addressing these challenges with the following: (A) Introduce REASONS, a dataset created by extracting related works from IEEE articles spanning 12 scientific domains from 2017 to 2023. (B) We employ a new RAG training regime to develop Advance RAG. Advance RAG and Na\u00efve RAG examine the factual integrity of the information retrieved by dense retrievers and its presentation as citations by LLMs. (C) We evaluate both proprietary and public LLMs and their RAG counterparts (10 models) to assess their contextual awareness using metrics like Pass Percentage (PP) and Hallucination rate (HR). Additionally, we have measured the quality of citation generation using F-1 and BLEU scores. (D) We conduct an adversarial examination to provide a clear assessment of context awareness regarding citation generation in LLMs. Findings:(I) Perplexity, faces a major challenge when dealing with indirect and direct query on the REASONS dataset (Figure 2 Figure 5, and in Appendix A Table 6 Table 9).(II) Citation generation is enhanced uniformly across public and proprietary LLMs when metadata like abstract and title are considered with indirect query (Figure 3 and Figure 5, along with Table 7 and Table 9). (III) Advance RAG with Mistral LLM outperforms other competitive proprietary and public LLMs. This performance is realized by a reduction in the HR and increments in F-1 and BLEU scores (Figure 3 and Figure 5 (last two bars) and Table 7 and Table 9 (last two columns)). (IV) For domains such as Quantum Computing and Biomolecules that are heavy in mathematics and numerals, there was a substantial decline in citation generation quality and an increase in HR. Adversarial examination strengthens our understanding that despite being exorbitantly large, LLMs lack context awareness (Table 2). (V) Advance RAG did provide convincing evidence of context understanding (Table 2). Further improvements in RAG-based LLMs are desirable, and utilizing REASONS dataset can provide valuable insights into context understanding and provenance in tasks such as hypothesis generation. 2 Background Early Techniques in Citation Recommendation: The practice of citing sources is a cornerstone of academic and professional writing, serving as the bedrock for reliability, and truthfulness in scholarly work (Cronin, 1981). The evolution of citation recommendation systems mirrors the broader advancements in computational linguistics and nat\fural language processing (NLP) (Bai et al., 2019; Ali et al., 2021). Initial methods in citation recommendation focused on basic techniques such as text feature-based systems (Strohman et al., 2007), simple keyword matching, and basic statistical methods (Bethard and Jurafsky, 2010). Context-aware citation recommendation systems supplemented these methods (He et al., 2010; Ebesu and Fang, 2017; Jeong et al., 2020a; Huang et al., 2021). However, their inability to grasp deeper textual contexts limited their effectiveness. Machine learning in Citation Recommendation The incorporation of machine learning into citation recommendation systems represents an initial step toward automating the citation process, which is typically regarded as manual and laborintensive(Agarwal et al., 2005; K\u00fc\u00e7\u00fcktun\u00e7 et al., 2012). These systems began to exhibit an improved understanding of the text, although they still lacked a nuanced grasp of complex contexts (Tran et al., 2015). The application of neural networks revolutionized citation recommendation. NLP algorithms, capable of parsing complex sentence structures, started identifying relevant themes for contextually appropriate citation recommendations (Zarrinkalam and Kahani, 2013; Beel et al., 2016; Iqbal et al., 2020). Concurrently, graph-based models, visualizing literature as interconnected networks, enhanced citation recommendations by considering content similarity and citation patterns (Ali et al., 2020; Chakraborty et al., 2015). With deep learning, citation recommendation systems began incorporating semantic analysis, employing models like word embeddings and neural networks for a more nuanced understanding (Yang et al., 2018; Bhagavatula et al., 2018; Vajdecka et al., 2023). Adapted from commercial use, collaborative filtering also emerged, recommending citations based on similar citation behaviors (Wang et al., 2020). Large Language Models in Citation Generation: The advent of LLMs like GPT-3 and its successors has further transformed NLP. Initial language model systems such as those based on BERT have significantly improved citation recommendation by converting unstructured text into meaningful vectors (Jeong et al., 2020b; Devlin et al., 2018; Bhowmick et al., 2021). Recent studies have focused on evaluating the fidelity of generated text to its sources (Ji et al., 2023). (Rashkin et al., 2023) introduced the \u201cattributable to identified sources\u201d (AIS) score, while (Bohnet et al., 2022) and others (Honovich et al., 2022; Yue et al., 2023) have focused on automating AIS. Concurrent work by (Liu et al., 2023) explored human evaluation of commercial generative search engines such as Bing. Chat, NeevaAI, Perplexity.ai, and YouChat. Despite these advancements, LLMs in citation recommendation still struggle with generating accurate information and providing references, as shown in studies by (Ji et al., 2023; Zheng et al., 2023). We conduct empirical and investigative research on why public and proprietary LLMs, including the powerful GPT-4 (which has not been examined yet), are prone to incorrect citation generation. Further, we provide means for improving the citation generation in public LLMs through a customized design using RAG. This limitation necessitates an approach closely aligning with RAG. RAG compels LLMs to provide citations alongside the generated text. The concept of retrieval-augmented LLMs has gained traction in recent years following (Guu et al., 2020; Borgeaud et al., 2022; Izacard et al., 2022; Khandelwal et al., 2019; Schick et al., 2023; Jiang et al., 2023b; Yao et al., 2022; Gao et al., 2023). We evaluate public and proprietary LLMs and their RAG counterparts on citation generation using REASONS, a meticulously curated dataset from arXiv spanning key domains in computer science and related fields. This allows us to assess the LLM\u2019s ability to identify a given sentence\u2019s source accurately. Domain Paper Count IEEE Papers Citation Count CV 5488 1028 3437 Robotics 3656 292 776 Graphics 1796 384 1417 IR 1741 564 1654 AI 1697 531 2021 NLP 1526 293 1092 Cryptography 1084 371 1106 NNC 892 111 326 HCI 761 112 229 Databases 723 115 182 QC 421 126 456 Biomolecules 119 17 27 Total 19904 3944 12723 Table 1: Our benchmark dataset, REASONS, includes papers and sentences from 12 domains. It primarily features ten domains in computer science and 2 in biology. Full forms of domain acronyms are provided in subsection A.5. \f3 Problem Setup Scope of REASONS: The dataset comprises sentences gathered from the related work sections of articles in computer science and biology available on arXiv (arX). Summary is provided in Table 1. It should be noted that GPT-3.5 or its successors may have processed all the papers published on arXiv from 2017 to 2021 while training. To ensure our dataset is unbiased, we include papers published in 2022 and 2023 that test the memory and understanding of LLMs. Exclusions were made for mathematics, statistics, and physics due to the abundance of equations in the related work section, and the crawling method theoremKb1 lacked the required versatility. We chose to focus on IEEE papers as they are represented across all 12 domains we considered. Each sentence in the related work section encapsulates the author\u2019s thought process in citing related works: (A) Every sentence captures the author\u2019s interpretation and emphasis on original methodology, critique of prior work, corrections to previous research, or acknowledgment of pioneers. This encompasses summarizing these aspects briefly and concisely. (B) The cited work in the related work section is either incidental or important to current work (Valenzuela et al., 2015). REASONS is inspired by previously constructed s2ORC and UnarXive datasets containing academic papers (see Table 4 in Appendix A); however, we diverge on the following points: (A) We provide sentence-level annotation of citations on major computational domains on arXiv. (B) Each sentence is accompanied by its metadata, which includes the paper title, abstract, and author names of the paper it cites. It also contains the title of the paper from which it was taken. (C) The dataset structure allows for an easy examination of LLMs using indirect and direct queries. Crawling Process: The web crawler employs the Oxylabs2 SERP Scraper API as its methodology, enabling real-time data extraction from major search engines. This API offers a proxy chaining platform for efficient data extraction. The dataset is meticulously organized in JSON format with a detailed outline (see \u201cJSON Structure\u201d). A complete GitHub repository is provided, containing the dataset and the code for reproducibility (see details in subsection A.3). We plan to keep updating the repository with more articles and metadata. The 1https://github.com/PierreSenellart/theoremkb 2https://oxylabs.io/ associated costs are provided in (subsection A.2). JSON Structure {\"Computer Vision\": { \"http://arXiv.org/abs/2012.05435v2\": { \"Paper Title\": \"Optimization-Inspired..\", \"Sentences\": [ {\"Sentence ID\": 32, \"Sentence\": \"... For GM, ... \", \"Citation Text\": \"C. Ledig,...\", \"Citation\": { \"Citation Paper ID\": \"arXiv:1609.04802\", \"Citation Paper Title\": \"Title:Photo..\", \"Citation Paper Abstract\": \"Abstract:.\", \"Citation Paper Authors\": \"Authors:...\" }}]}}} 3.1 Problem Formulation We define two tasks for LLMs over the REASONS dataset R: (a) Direct Querying and (b) Indirect Querying. For experimentation, we segment R into RS and RM. RS represents sentences and paper titles for which references are to be generated with or without the support from metadata RM. Direct Querying Task: Given a title ti \u2208RS, the LLM should generate the author list. For the task of direct querying with metadata, the LLM is given the following input: ti \u2208RS, the Advance RAG model retrieves top-40 chunks of information ai1, ..., ai40 \u2208RM, and generates the names. Indirect Querying Task: Given a sentence si \u2208RS, the LLM should generate a paper title in zero-shot setting. For the task of indirect querying with metadata called Sequential Indirect and Direct Prompting (SID Prompting), the LLM is given the following input: si \u2208RS and ground truth abstract abss \u2208RM as well as the authors aus \u2208RM, and the model is asked to generate the citation paper title. Examples of direct and indirect queries are: Direct Prompt Prompt: Who were the authors of the research paper \"Research Paper Title\"? Instruction: List only author names, formatted as < firstname >< lastname >, separated by comma. Do not mention the paper in the title, also, if you don\u2019t know, write \u2019pass\u2019. Response: Author Names. \fIndirect Prompt Prompt: I have taken a sentence from the research paper titled \u201cResearch Paper Title\u201d, give me the research paper that this sentence is citing. If you cannot come up with the paper titles, write \u2018pass.\u2019 Don\u2019t write anything else. Instruction: Sentence \"uses fractional max-pooling to randomly specify non-integer ratios between the spatial dimension sizes of the input and the output to pooling layers.\" Response: Citation Paper Title. Implementation of Direct and Indirect Querying: Direct querying is executed using zero-shot prompting for scenarios without metadata and chain-of-thoughts prompting for metadata situations. We modify the chain-of-thoughts prompting with SID Prompting. It begins with an indirect query. Following an incorrect response or a \u2018pass,\u2019 more details about the cited paper are given (i.e., direct query), including its abstract and authors\u2019 names. This is an iterative approach to generate the correct citation. Following are the two examples of these prompting strategies: Direct Query with Metadata Prompting Prompt: Who were the authors of the research paper \u201cResearch Paper Title\"? Let me give you some more context by providing the abstract of the research paper. Abstract:\u2019....\u2019. Instruction: List only author names, formatted as <first name><last name>, separated by comma. Do not mention the paper in the title. Also, if you don\u2019t know, write \u2018pass.\u2019 Response: Author Names. SID Prompting Prompt: I have taken a sentence from the research paper titled \"Research Paper Title.\" give me the title of the possible research paper that this sentence is citing. If you cannot come up with the paper titles, write \u2019pass\u2019. Don\u2019t write anything else. Instruction: Sentence:\"......\". Let me give you some more context by providing the authors and the abstract of the paper the sentence is citing. Authors:\"......\", Abstract:\".......\" Response: Citation Paper Title. 3.2 Models and Evaluation Our research has focused on a diverse array of LLMs, carefully chosen to provide a broad perspective on the capabilities and limitations inherent in current language model technologies. Proprietary Models: Our selection of proprietary models includes those from OpenAI and Preplexity.ai. While OpenAI is known for its cutting-edge NLP models, driving significant advancements in the field, Preplexity.ai focuses on models with unique functionalities, such as recommending citations and utilizing natural language prediction for innovative search experiences. Public Models: We choose LLAMA 2 (Touvron et al., 2023) and Mistral (Jiang et al., 2023a) as the two publicly available LLMs that have demonstrated competitive performance compared to proprietary LLMs. We evaluate their effectiveness on the REASONS dataset under the standard state and retrieval-augmentation conditions. This analysis goes beyond simply comparing proprietary and public models, extending to evaluating models based on their size, particularly those with 7B parameters. 3.3 Evaluation Metrics Our evaluation uses four key metrics: 1) The BLEU Score assesses the structural alignment through clipped n-gram matching. 2) The F-1 Score evaluates the balance between precision and recall, reflecting the models\u2019 effectiveness in capturing key information. 3) Hallucination rate (HR), which we estimate by averaging over incorrect and partially correct generated citations. HR = 1 QD P I[\u02c6 c \u0338= c] + 1 |Uw| P|Uw| w=1 I[\u02c6 cw \u0338= cw], where QD: queries within a domain, and |Uw|: total number of unique words in generated citation (\u02c6 c) and true citation (c). 4) Pass Percentage (PP) measures the tendency of an LLM to either respond or abstain from giving a response. It is calculated as follows: 1 QD P I[\u02c6 c = Pass]. It is crucial to emphasize that PP serves as a safeguard to prevent LLMs from generating hallucinatory responses but also reduces engagement. Additionally, even with a high PP, the HR can be high. This implies that the model struggles to discern whether it offers correct or incorrect citations in the remaining instances. 3.4 Retrieval Augmented Generation (RAG) RAG combines a retriever and a generator to create better answers. RAG can access external knowledge, unlike methods that feed the model prompts. This lets it craft more accurate, relevant, and informative responses than models that rely solely on what they were pre-trained. We investigate RAG\u2019s ability to improve LLMs\u2019 accuracy. Ideally, RAG would help LLMs avoid giving wrong answers (low PP) and making things up (HR). We also investigate whether RAG works consistently with direct and indirect questions across different scientific fields (12 domains). We experiment with two forms of RAG architecture: \f(a) Na\u00efve RAG and (b) Advance RAG. Both architectures leverage the same bi-encoder-based retriever architecture (Karpukhin et al., 2020). Given a corpus of documents RM and a sentence s \u2208RS, the document encoder maps d \u2208RM to an embedding E\u03b8(c) and the query encoder maps s to an embedding E\u03b8(s). The top-k relevant documents for s are retrieved based on the sentence-document embedding similarity, which is often computed via dot product: z(s, d) = exp(E\u03b8(s)T E\u03b8(d)). We start with a bi-encoder retriever using an embedding model from OpenAI (subsection A.4). Other ways to set up a bi-encoder retriever, such as DRAGON+ (Lin et al., 2023), are possible. However, those are more useful when involving large-scale data augmentation. The retrieved documents are ranked in two ways, which separates Na\u00efve RAG from Advance RAG. Under the Na\u00efve RAG, we use BM25 relevance scoring to rank the documents, whereas, in Advance RAG, we fine-tune a cross-encoder on REASONS document index RM to better align it with our task of citation generation with LLM. For the fine-tuning of the cross-encoder, we use localized contrastive loss (LCL) for two reasons: (a) In RM, we do not have labeled positive and negative documents, and (b) for a sentence s there is a possibility for more than one true positive documents (Pradeep et al., 2022). LCL is formally defined as follows: LLCLs := \u2212log exp(zs,{d+}) P d\u2208Gs exp(zs,d) LLCL := 1 |S| X s\u2208Rs,Gs\u2208Rs M LLCLs where Gs represents a set of documents for a sentence s, which consist of a set of relevant documents ({d+}) and n-1 non-relevant documents {d\u2212} sampled from Rs M using biencoder. The training of Advance RAG happens through the standard cross entropy loss: LCE(\u02c6 c|s, \u03d5) = Pb i=1 I(\u02c6 cw i = cw i ) \u00b7 log Pr(\u02c6 cw i |\u03d5) where, \u03d5 is parameter of the generator LLM and b is the minibatch fine-tuning in Advance RAG. \u02c6 ci represents ith citation generation, and I(\u02c6 cw i = cw i ) represents word level comparison with ground truth citation (direct query: author names; indirect query: paper titles). For the Na\u00efve and Advance RAG, we employ LLAMA-2 7B and Mistral 7B as competitive models against proprietary LLMs. 4 Results We conducted experiments encompassing four distinct prompting styles applied to twelve scientific domains. This extensive analysis involved 12,723 sentences, resulting in a substantial dataset rigorously evaluated using ten different models. This equates to 508920 instance assessments involving 4 (prompting styles) \u00d7 12,723 (sentences for all domains) \u00d7 10 (models). The total duration required to execute all experiments on the GPU is 238 days, 6 hours, and 59 minutes. For detailed information regarding the time spent on experiments across various domains, please refer to the appendix (see subsection A.6 and Table 5). Zero-Shot Indirect Prompting: In Figure 4, a majority of the models exhibited high HR. As expected for a huge model GPT-4-1106-preview (1 Trillion Parameters) shows a relatively lower HR of 67.73% and a higher PP of 89% averaged across 12 domains. Perplexity-7b-Chat showed an exceptionally high PP of 97.5%, which is surprising, as this LLM is designed specifically for citation generation. RAG Mistral was a competitive model with GPT-4 with a lower PP of 21% and HR of 72.49% in comparison to other LLMs. Analysis shows RAG Mistral is competitive because of the high variance in HR compared to GPT-4-1106-preview. Generation quality measured by F-1 and BLEU scores were predominantly low across the board, with GPT-4 (not the preview, G1) comparatively better scores. RAG Mistral and RAG LLAMA 2 rank second and third best respectively. SID Prompting In Figure 5, showed improvement across all the LLMs in citation generation over indirect queries. An average improvement of 21% was measured, with a reduction in variance. Even though some models like Perplexity-7b-Chat and LLAMA 2 still had high HR rates, the PP dropped significantly, especially for GPT-4-1106-preview. The results of this experiment indicate that SID prompting in LLMs can balance the trade-off between PP and HR, significantly enhancing generation quality with an (8%\u2191) increase in BLEU and a (13%\u2191) in F-1 (The Appendix B provides examples for visual inspection.). Zero-Shot Direct Prompting presents a very idealistic scenario where the LLMs have access to context through direct query. This leads to both lower PP and HR. The citation generation quality significantly improves from zero-shot in\fG1 G2 G3 P RMM RL L AL AM 0 50 100 Hallucination Rate G1 G2 G3 P RMM RL L AL AM 0 0.2 0.4 0.6 0.8 F-1 Score G1 G2 G3 P RMM RL L AL AM 0 0.2 0.4 0.6 0.8 BLEU Score G1 G2 G3 P RMM RL L AL AM 0 50 100 Pass Percentage Figure 2: Averaged Zero-Shot Direct Prompting results of different LLMs across all 12 domains. G1 shows notably lower HR and higher F-1 and BLEU scores, indicating superior performance in generating citations. In contrast, model P exhibits the highest HR and the lowest scores in F-1 and BLEU, suggesting challenges in generating accurate and contextually relevant citations. The RAG models (RM and RL) demonstrate varied results, with RM showing a better accuracy and coherence balance than RL. G1: gpt-4-1106-preview, G2: gpt-4, G3: gpt-3.5-turbo, P: pplx-7b-chat, RM: Na\u00efve RAG mistral-7b-instruct, M: mistral-7b-instruct, RL: Na\u00efve RAG llama-2-7b-chat, L: llama-2-7b-chat, AL: Advance RAG llama-2-7b-chat, AM: Advance RAG mistral-7b-instruct. For the purposes of clarity and saving space, the terms AL and AM are used in the figures to denote Advance RAG llama-2-7b-chat and Advance RAG mistral-7b-instruct, respectively. In the main text of the paper, these are referred to as AdvRAG(L) and AdvRAG(M). G1 G2 G3 P RMM RL L AL AM 0 50 100 Hallucination Rate G1 G2 G3 P RMM RL L AL AM 0 0.5 1 F-1 Score G1 G2 G3 P RMM RL L AL AM 0 0.5 1 BLEU Score G1 G2 G3 P RMM RL L AL AM 0 0.5 1 Pass Percentage Figure 3: Averaged Direct Prompting with Metadata results of different LLMs across all 12 domains. The plot indicates that models G1, G2, and G3 stand out with their low HR and impressive F-1 and BLEU scores, in contrast to other models that face challenges. All models except RM reach a 0% PP, suggesting that including metadata significantly enhances their contextual understanding. G1 G2 G3 P RM M RL L 0 50 100 Hallucination Rate G1 G2 G3 P RM M RL L 0 0.2 0.4 0.6 F-1 Score G1 G2 G3 P RM M RL L 0 0.2 0.4 0.6 BLEU Score G1 G2 G3 P RM M RL L 0 50 100 Pass Percentage Figure 4: Averaged Zero-Shot Indirect Prompting across 12 domains. This prompting method led to elevated HR among the models. There was also a notable variance in PP, with models G3, P, and L exhibiting higher scores. Both conditions indicate challenges in understanding context and generating accurate citations when using indirect prompts. G1 G2 G3 P RMM RL L AL AM 0 50 100 Hallucination Rate G1 G2 G3 P RMM RL L AL AM 0 0.2 0.4 0.6 0.8 F-1 Score G1 G2 G3 P RMM RL L AL AM 0 0.2 0.4 0.6 0.8 BLEU Score G1 G2 G3 P RMM RL L AL AM 0 50 100 Pass Percentage Figure 5: Averaged SID Prompting results of different LLMs across all 12 domains. Models G1, G2, and G3 exhibit relatively better outcomes with lower HR and higher F-1 and BLEU scores, suggesting more contextual understanding. Other models demonstrated high HR, indicating difficulties in accurate citation generation with SID Prompting. Notably, while models G1 and G3 have high PPs, indicating some difficulties with SID, their overall performance still reflects a more advanced level of language processing and contextual comprehension compared to the other models. direct and SID promptings, achieving high F-1 and BLEU scores (see Figure Figure 4). However, Perplexity-7b-Chat, oddly, had high PP and HR, suggesting a need for more research on such \fspecialized LLM search engines. We observed that Perplexity-7b-Chat expands its search queries and adds references to the broader content it finds. The issue is that the expanded versions drift too far in meaning from the original. In Direct Prompting with Metadata, when metadata such as abstracts and titles were used with indirect questions, all the LLMs got better at generating citations and had low HR and PP. This shows that having more information helps LLMs create more accurate and related citations, proving the importance of enough data for good language processing. Note that PP dropped to zero for almost all models when direct promoting includes metadata. All GPT LLMs achieved F-1 and BLEU scores close to 1.0 and showed more consistent results overall. Two main points from this experiment are: First, adding metadata to LLMs is effective for all of them, especially RAG models that integrate this augmentation in their learning process. Second, smaller models with advance RAG (Mistral and LLAMA-2) adjust better to metadata than GPT-4-Preview/4/3.5 (see Figure 3). Overall: Advance RAG Mistral 7b outperformed other competitive proprietary and public LLMs in all prompting styles. This superior performance was notably marked by reduced HR, suggesting this model is more adept at generating accurate and relevant responses when adding metadata. Furthermore, improvements in F-1 scores reinforce its reliability in retrieving information. Higher BLEU scores were observed, signifying that the language output of the model aligns closely with human-like text in terms of fluency & coherence. 5 Adversarial Examination The analysis of LLMs using the REASONS dataset highlights significant variability in their performance across different domains. While they perform moderately better in areas like AI and CV with lower HR and higher F-1/BLEU scores, they struggle in complex domains such as QC, Biomolecules, and Cryptography, likely due to limited training data and the complexity of these subjects. This variability in performance indicates that LLMs have varying degrees of contextual understanding, with a tendency to perform better in domains with more extensive training data and less complex structures (e.g., maths and numerics). Motivation and Setup: We conducted adversarial experiments across all models to better assess their contextual understanding. The core concept Group PP(%) BLEU F1 HR Changing Paper Title G1 96.23 0.6210 0.8470 17.99 G2 31.45 0.0524 0.2640 83.66 G3 68.55 0.0389 0.1828 87.35 RM 3.14 0.0796 0.1584 86.78 M 0.00 0.0003 0.0221 94.95 RL 5.03 0.0628 0.1448 87.56 L 0.00 0.0066 0.0254 98.30 AdvRAG(L) 0.00 0.1322 0.4763 85.72 AdvRAG(M) 0.00 0.1569 0.5839 75.41 Changing Paper Abstract G1 95.60 0.4595 0.6451 38.49 G2 32.70 0.0396 0.2186 86.22 G3 76.10 0.0034 0.1013 91.64 RM 7.55 0.0520 0.1216 89.44 M 0.00 0.0074 0.0161 90.20 RL 2.52 0.0445 0.1112 90.16 L 0.00 0.0017 0.0146 99.01 AdvRAG(L) 0.00 0.4101 0.5780 39.67 AdvRAG(M) 0.00 0.4904 0.6954 39.57 Table 2: Performance of various LLMs on adversarial set, designed by swapping titles and abstracts. Models G1, G2, and G3, possibly exposed to similar data during training, struggled with the adversarial sets, resulting in high HR and PP. Conversely, models like AdvRAG(L) and AdvRAG(M) showed better performance, suggesting that these models attempt to understand the context before generating the citations. behind these experiments was to provide the models with incorrect yet similar metadata about the sentences in the prompts. The aim was to discern whether the models generated citations based on the contextual grasp of the provided metadata or if the metadata had minimal influence on the citation generation process. These adversarial experiments comprised two types: 1) Providing inaccurate paper titles related to the sentences. 2) Providing incorrect paper abstracts associated with the sentences. Both experiments were conducted using the SID prompting. To facilitate these experiments, we curated a subsample of 200 sentences from the REASONS dataset spanning all the domains. We extracted each sentence\u2019s most similar paper title or abstract from this dataset and replaced the original metadata. For similarity calculation, we use the RatcliffObershelp metric, which is calculated as twice the length of the longest common substring plus recursively the number of matching characters in the non-matching regions on both sides of the longest common substring (Tang et al., 2023). According to this metric, for the following example title \u201cDiffusion models for counterfactual explanations,\u201d the best replacement is \u201cOctet: Object-aware models for counterfactual explanations (0.736)\u201d as opposed \fto \u201cAdversarial counterfactual visual explanations (0.638)\u201d. We considered a threshold of 0.70 effective in preparing the adversarial set. Findings: We found that incorrect paper titles and abstracts easily fool most LLMs if it is similar to accurate information. In Table 2, G1 is displayed at 17.99%, and its pairing with a high PP of 96.23% indicates a defensive mechanism. This means the LLMs are not very good at understanding the true meaning of what they are given. On such a small adversarial set, we expect LLMs like GPT-4-1106-preview and GPT-4 to perform exceedingly well because of their extensive knowledge; however, we observed counterintuitive results in Table 2, all models show the effect. We do see promising direction with AdvRAG(M) and AdvRAG(L); however, further investigation is required into how rich graphical metadata (e.g., knowledge graph) and graph-theoretic approaches to information retrieval can improve LLM effectiveness (He et al., 2024). 6 Conclusion We have developed a new resource called REASONS (REtrieval and Automated citationS Of scieNtific Sentences), a benchmark designed to assess the ability of LLMs to understand context and generate appropriate citations. This benchmark includes sentences from the related work sections of papers, along with citations and metadata across 12 scientific and computational fields. We evaluated proprietary and public LLMs\u2019 ability to correctly provide author names and paper titles under two conditions: direct and indirect citation. Surprisingly, none of the LLMs demonstrated the readiness to annotate draft reports in various professional settings, such as market analysis, misinformation prevention, defense strategy, and healthcare reporting. We observed a trade-off between PP and HR, where GPT-4 and GPT-3.5 achieved higher accuracy at the cost of a lower HR. In contrast, though smaller with only 7B parameters, the Advance RAG model showed reasonable efficiency. Unlike other models, in adversarial tests where abstracts or paper titles were swapped, Advance RAG unexpectedly outperformed GPT-4, suggesting it does capture context before generating citations. Future Work: Through reasoning and explanation, we plan to explore and mitigate the noted shortcomings in citation generation (trade-off between HR and PP, high variance in BLEU scores, sub-par scores on adversarial set). One approach is to employ the Toulmin model (Naveed et al., 2018)) within Advance RAG. We believe these improvements will improve the quality of citation generation and better equip the models to manage complex reasoning (e.g., hypothesis generation and verification (Tyagin and Safro, 2023)) challenges confidently. Limitations Several factors constrain our study on applying LLMs for citation generation. (a) Primarily, integrating high-parameter-size models (>13B; refer to Table 5 for computation time) with RAG is not feasible, limiting our ability to leverage more complex models. (b) Additionally, the high computational resources required for such models are often inaccessible in academic settings. (c) One constraint in our study was the dataset creation, where we confined ourselves to predominantly IEEE format papers, particularly with domains with a high count of submissions. (d) Another significant limitation is the current inability of LLMs to effectively process and interpret mathematical expressions, a crucial aspect in many academic papers. (e) Due to the latest version of Google API (time stamp: December 04, 2023) lacking the citation generation feature, we have limited our experiments to OpenAI only. (f) While cross-encoders can be more powerful in understanding text relationships, they tend to be more computationally intensive. This is because they need to process every possible pair of inputs together, which can be a significant workload, especially in cases where there are many potential pairs to consider (like in large-scale retrieval tasks in our REASONS dataset). These constraints highlight the need for advancements in model adaptability, computational resource accessibility, dataset diversity, and specialized content processing for more robust and wide-ranging applications. Ethical Considerations We followed the Oxylabs Acceptable Use Policy3 and worked alongside some Oxylabs developers to ensure we respected the terms of services on arXiv. arXiv\u2019s terms of service place restrictions on automated crawling of their site for articles marked by \u201carxiv.org perpetual, non-exclusive license and CC BY-NC-ND\u201d. We paid attention to the following key ethical issues: (a) Privacy and Consent: The content on arXiv is publicly available, but the authors who upload their work there may not have 3https://oxylabs.io/legal/ oxylabs-acceptable-use-policy \fconsented to having their preprints crawled and used for other purposes. It\u2019s important to respect the privacy and intellectual property rights of the researchers who contribute to arXiv. We only crawled articles marked as CC Zero, CC BY, and CC BYSA. (b) Potential misuse: We prepared REASONS only to test the citation generation capability of LLMs for subsequent future downstream applications, such as annotating draft analytic reports. Our focus on HR and PP for citation generation and its quality using BLEU and F-1 shows that the data scraped is not for malicious purposes, such as fine-tuning LLMs to generate misinformation or infringe on copyrights. (c) Transparency and Accountability: We have been mindful of our crawling process, and to the best of our knowledge, we have enumerated sufficient details regarding the process. This would help build trust regarding reproducibility, extend REASONS, and ensure that the crawling process was not abused. (d) Author Identity and Contact: No authors of the crawled papers were contacted through their provided information in the publicly available arXiv papers. This user study was duly approved by the authors\u2019 organization\u2019s Institutional Review Board (IRB).",
+    "additional_graph_info": {
+        "graph": [
+            [
+                "Ali Mohammadi",
+                "Thang Pham"
+            ],
+            [
+                "Ali Mohammadi",
+                "Sophie Stevens"
+            ],
+            [
+                "Ali Mohammadi",
+                "Alina Ostafe"
+            ]
+        ],
+        "node_feat": {
+            "Ali Mohammadi": [
+                {
+                    "url": "http://arxiv.org/abs/2111.04072v2",
+                    "title": "A Point-Conic Incidence Bound and Applications over $\\mathbb F_p$",
+                    "abstract": "In this paper, we prove the first incidence bound for points and conics over\nprime fields. As applications, we prove new results on expansion of bivariate\npolynomial images and on certain variations of distinct distances problems.\nThese include new lower bounds on the number of pinned algebraic distances as\nwell as improvements of results of Koh and Sun (2014) and Shparlinski (2006) on\nthe size of the distance set formed by two large subsets of finite dimensional\nvector spaces over finite fields. We also prove a variant of Beck's theorem for\nconics.",
+                    "authors": "Ali Mohammadi, Thang Pham, Audie Warren",
+                    "published": "2021-11-07",
+                    "updated": "2022-07-25",
+                    "primary_cat": "math.CO",
+                    "cats": [
+                        "math.CO"
+                    ],
+                    "main_content": "Introduction For an arbitrary \ufb01eld F and sets of points P and algebraic curves C in Fd, we denote the number of incidences between P and C by I(P, C) := |{(p, C) \u2208P \u00d7 C : p \u2208C}|. The celebrated Szemer\u00b4 edi\u2013 Trotter theorem [26] states that for \ufb01nite sets of points P and lines L over R2, one has I(P, L) \u226a|P| 2 3 |L| 2 3 + |P| + |L|. (1.1) It is well-known that this bound is sharp up to a constant factor. Many applications of this result can be found in [8, 28], and references therein. Let Fq denote a \ufb01nite \ufb01eld of order q and characteristic p. A \ufb01nite \ufb01eld analogue of the Szemer\u00b4 edi\u2013 Trotter theorem was \ufb01rst studied by Bourgain, Katz and Tao [5] in 2004. They proved that for any point set P and any line set L in F2 p with |P| = |L| = N = p\u03b1, 0 < \u03b1 < 2, we have I(P, L) \u226aN 3 2\u2212\u03b5, where \u03b5 = \u03b5(\u03b1) > 0. (1.2) The study of this type of incidence structure is not only interesting from a geometric perspective, but is also largely motivated by a wide range of applications in di\ufb00erent areas, such as arithmetic combinatorics, number theory, restriction theory, and theoretical computer science. Since its appearance nearly two decades ago, only a few quantitative variants of estimate (1.2) have been proved. In particular, when N is large with respect to the order of the \ufb01eld, say N \u2265p, Vinh [30] showed that \f \f \f \fI(P, L) \u2212|P||L| p \f \f \f \f \u2264p1/2p |P||L|. (1.3) This was proved using techniques from graph theory. A very short and elementary proof can \u2217School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran. Email: a.mohammadi@ipm.ir \u2020Department of Mathematics, HUS, Vietnam National University. Email: thangpham.math@vnu.edu.vn \u2021Johann Radon Institute for Computational and Applied Mathematics (RICAM), Linz, Austria. Email: audie.warren@oeaw.ac.at 1 \falso be found in [17]. This result also holds for the setting of arbitrary \ufb01nite \ufb01elds Fq. When |P| = |L| = p3/2, it follows from (1.3) that I(P, L) \u223cN 4/3, which matches the bound of (1.1). On the other hand, it has been indicated in [31] that the lower bound of (1.3) is sharp in the sense that there are sets P and L with |P||L| \u226bp3 and there is no incidence between P and L. When N is extremely small, say, N \u226alog2 log6 log18 p, Grosu [9] proved that the point-line incidence structure in F2 p is almost the same as that in C2. As a consequence, relying on the C2 analogue of estimate (1.1) due to T\u00b4 oth [29], he obtained I(P, L) \u226aN 4/3. Therefore, we are left with the situation that log2 log6 log18 p \u226aN \u226ap3/2. In the case |P| = |L| = N \u2264p, Helfgott and Rudnev [10] proved that I(P, L) \u226aN 3 2 \u2212 1 10678 . Jones [13] showed that the saving of 1 10678 can be improved to 1 662. Both results in [10, 13] were proved by using sum-product type energy inequalities in the \ufb01eld Fp. The best current result is due to Stevens and de Zeeuw [25], who proved that for any set P of m points in F2 p and any set L of n lines in F2 p, if m7/8 < n < m8/7 and m\u22122n13 \u226ap15, then the number of incidences between P and L satis\ufb01es I(P, L) \u226am11/15n11/15. Their proof relies mainly on Rudnev\u2019s point-plane incidence bound [21]. We point out that this theorem also holds in the setting of arbitrary \ufb01elds F. In this case, the condition m\u22122n13 \u226ap15 is replaced by m\u22122n13 \u226achar(F)15, and is removed if the characteristic of F is zero. In a very recent paper, Rudnev and Wheeler [22] obtained point-hyperbola incidence bounds, that is, incidences between points and hyperbolas of the form (x \u2212a)(y \u2212b) = 1 in F2 p, which improve earlier results due to Shkredov in [23] and Bourgain in [4]. The main idea in their argument is as follows: for a \ufb01xed point q \u2208P, let Hq be the set of hyperbolas passing through q. They observed that the number of k-rich hyperbolas in Hq can be estimated by using the Stevens\u2013de Zeeuw point-line incidence bound. This observation has been used by Warren and Wheeler [32] to derive an incidence bound between a point set and a set of M\u00a8 obius transformations in the plane F2 p. The main purpose of this paper is to employ this key idea to study a general incidence problem, namely, incidences between points and irreducible conics in the plane F2 p. We recall that the three types of irreducible conics include parabolas, ellipses and hyperbolas. We also remark that the results in Sections 1.1 and 1.2 can be extended to arbitrary \ufb01elds, as long as the relevant characteristic condition is satis\ufb01ed this is because each of these results relies on the point line incidence bounds of Stevens and de Zeeuw, which hold over arbitrary \ufb01elds. The results of Warren and Wheeler also rely on these results, and can therefore also be extended to arbitrary \ufb01elds. In Section 6, we give various applications of our incidence results. For instance, we prove a pinned algebraic distances result, and an analogue of Beck\u2019s theorem for conics in \ufb01nite \ufb01elds. 1.1 Point-conic incidence bounds over prime \ufb01elds Our \ufb01rst result gives an upper bound on the number of incidences between small sets of points and irreducible conics over F2 p. Theorem 1.1. For any set C of irreducible conics in F2 p, and any set of points P \u2286F2 p with |P| \u226ap15/13, we have I(P, C) \u226a|P|23/27|C|23/27 + |P|13/9|C|12/27 + |C|. To compare with the Cauchy-Schwarz bounds, we note that any conic is determined uniquely by \ufb01ve points, no three of which are collinear, (see [1, Exercise 69]) and by B\u00b4 ezout\u2019s theorem, any two distinct conics meet in at most four distinct points. So by the K\u02dd ov\u00b4 ari\u2013S\u00b4 os\u2013Tur\u00b4 an theorem (see [3, 2 \fTheorem IV.10]), we have I(P, C) \u226amin{|P||C|4/5 + |C|, |P|1/2|C| + |P|}. (1.4) Theorem 1.1 improves the trivial bounds of (1.4), in the range |P|19/8 \u2264|C| \u2264|P|20/7, which encompasses the \u2018balanced Cartesian product\u2019 range that is, when the set of conics C have coe\ufb03cients (a, b, c, d, e) coming from a Cartesian product A \u00d7 B \u00d7 C \u00d7 D \u00d7 E, and the point set P is also a Cartesian product F \u00d7 G, and all the sets involved are of the same size N. In this case we have |C| = N 5 = |P|5/2, and our bound improves on (1.4). Next, we give an improvement of Theorem 1.1 for the particular case when our point set is a Cartesian product. Theorem 1.2. Let C be a set of irreducible conics in F2 p. Given any sets A, B \u2282Fp with |A| \u2264|B| and |A||C| \u226ap2, we have I(A \u00d7 B, C) \u226a|A| 3 4 |B| 5 8|C| 7 8 + |A| 1 2 |B| 3 4|C| 1 4 + |C|. In order to compare these results to what is known over the real numbers, the best analogue is given by the Pach-Sharir theorem [20], which implies that the number of incidences between a point set P and an arbitrary set of conics C satis\ufb01es I(P, C) \u226a|P|5/9|C|8/9 + |P| + |C|. We refer the reader to [27] for a survey of incidence results and their applications over Rd. 1.2 Point-circle, point-parabola and point-hyperbola incidence bounds over prime \ufb01elds When C is a set of circles, parabolas or hyperbolas, we have the following improvements. Theorem 1.3. Let C be either a set of circles, or of parabolas of the form y = ax2 + bx + c, or of hyperbolas of the form (x \u2212a)(y \u2212b) = c and P \u2286F2 p, with |P| \u226ap15/13. If C is a set of circles suppose that p \u22613 (mod 4). Then we have I(P, C) \u226a|P|15/19|C|15/19 + |P|23/19|C|4/19 + |C|. We remark that any two parabolas or circles meet in at most two points, and that they are determined uniquely by three non-collinear points, as a consequence of the K\u02dd ov\u00b4 ari\u2013S\u00b4 os\u2013Tur\u00b4 an theorem, we have I(P, C) \u226amin{|P||C|2/3 + |C|, |P|1/2|C| + |P|}. (1.5) Theorem 1.3 is better than this Cauchy-Schwarz bound in the range |P|11/8 \u226a|C| \u226a|P|12/7, again encompassing the balanced Cartesian product range analogous to that described above. As before, an improved estimate is obtained when P is a Cartesian product. Theorem 1.4. Let A, B \u2282Fp, and let C be either a set of circles, or of parabolas of the form y = ax2 + bx + c, or of hyperbolas of the form (x \u2212a)(y \u2212b) = c. If C is a set of circles, suppose that p \u22613 (mod 4). If |A||C| \u226ap2, then we have I(A \u00d7 B, C) \u226a|A|4/5|B|3/5|C|4/5 + |A|6/5|B|7/5|C|1/5 + |C|. 3 \f1.3 Incidence bounds over arbitrary \ufb01nite \ufb01elds for large sets Using the same approach, we are able to extend those theorems into arbitrary \ufb01nite \ufb01elds for large sets as follows. Theorem 1.5. Let C be a set of irreducible conics in F2 q and P be a set of points in F2 q. We have I(P, C) \u226a|P||C| q + q1/5|P|4/5|C|4/5 + |C|. Theorem 1.6. Let P and S be sets of points and spheres in Fd q respectively and assume q \u22613 (mod 4). We have I(P, S) \u226a|P||S| q + q d\u22121 3 |P|2/3|S|2/3. (1.6) It is worth noting that Theorem 1.6 improves earlier results in the literature, namely, it is better than the bound |P||S| q + q d 2 (|P||S|)1/2 due to Cilleruelo, Iosevich, Lund, Roche-Newton and Rudnev in [6] when |P||S| \u2264qd+2, and is better than the bound |P||S| q + q d\u22121 2 (|P||S|)1/2, due to Koh, Lee and Pham [14], which holds for small sets S with |P||S| \u2264qd\u22121. In the range |P||S| \u2265qd+2, Theorem 1.6 implies the same bound as the above-mentioned result of Cilleruelo et al. in [6], which is optimal. 2 Proof of the point-conic bound (Theorem 1.1) A M\u00a8 obius transformation (over Fq) is a map f of the form f(x) = ax + b cx + d, ad \u2212bc \u0338= 0. We will freely swap between the notion of a M\u00a8 obius transformation as a map f, and the curve given by y = f(x). We require the following result of [32] on the number of k-rich M\u00a8 obius transformations, which are de\ufb01ned as Tk := {f \u2208T : |f \u2229P| \u2265k}, where T is a set of M\u00a8 obius transformations. Theorem 2.1. For any set T of M\u00a8 obius transformations, and any set of points P \u2286F2 p with |P| \u226ap15/13, for all k \u22653 we have |Tk| \u226a|P|15/4 k19/4 + |P|2 k2 . Proof of Theorem 1.1. We shall \ufb01rst aim to bound the number of k-rich conics in C, the set of which we denote by Ck. Fix two distinct points q1 and q2 in P. De\ufb01ne the set Cq1,q2,k := {C \u2208C : q1, q2 \u2208C, |C \u2229P| \u2265k}. 4 \fCq1,q2,k is a set of conics which all pass through the two points q1 and q2. We apply a projective transformation \u03c0, with the property that \u03c0 maps q1 and q2 as follows q1 \u2192[0 : 1 : 0], q2 \u2192[1 : 0 : 0]. Note that such a transformation always exists. We now analyse the image \u03c0(C) for each C \u2208 Cq1,q2,k. Since \u03c0 is a projective transformation, these images must all remain degree two algebraic curves, that is, conics. Furthermore, we know that the two points at in\ufb01nity [0 : 1 : 0] and [1 : 0 : 0] both lie on \u03c0(C). We claim that this in fact forces \u03c0(C) to be a M\u00a8 obius transformation. Indeed, letting \u03b3 denote an irreducible conic over Fp, we have \u03b3 is a M\u00a8 obius transformation \u21d0 \u21d2 {[0 : 1 : 0], [1 : 0 : 0]} \u2286\u03b3. (2.1) First, we show the implication = \u21d2of (2.1). If \u03b3 is M\u00a8 obius, then since it corresponds to a conic, it is given by an equation of the form y = ax + b x + c which we then rearrange to xy + cy \u2212ax \u2212b = 0. Projectivising this curve, we have xy + cyz \u2212axz \u2212bz2 = 0. Upon setting z = 0 to \ufb01nd the points at in\ufb01nity, we have xy = 0, and so one of x, y must be zero. This yields the two points as in (2.1). We now show the reverse implication. Projectivising an arbitrary conic, of the form ax2 + by2 + xy + cy + dx + e = 0, we have ax2 + by2 + xy + cyz + dxz + ez2 = 0. Setting z = 0 to \ufb01nd the points at in\ufb01nity, we have the equation ax2 + by2 + xy = 0. We know that the points (0, 1) and (1, 0) must lie on this curve. The \ufb01rst implies that b = 0, and the second that a = 0. Therefore the original curve \u03b3 is of the form xy + cy + dx + e = 0. We also must have dc \u0338= e, as otherwise this conic is reducible as (y + d)(c + x) = 0. Therefore, we have shown that \u03b3 is the general form of a M\u00a8 obius transformation, concluding the proof of (2.1). With (2.1) at hand, we know that the set \u03c0(Cq1,q2,k) is a set of M\u00a8 obius transformations, and so we can apply Theorem 2.1. Therefore, for each distinct pair (q1, q2) \u2208P2, and for each k \u22655, we have |Cq1,q2,k| \u226a|P|15/4 k19/4 + |P|2 k2 . (2.2) Note that the condition k \u22653 has changed to k \u22655, since the two points at in\ufb01nity on these conics are now being ignored. In order to alter this bound into a bound on Ck, we sum over each 5 \fdistinct pairs of points. |Ck| \u2264 \u0012k 2 \u0013\u22121 X q1,q2\u2208P |Cq1,q2,k| \u226a1 k2 X q1,q2\u2208P |Cq1,q2,k|. The binomial factor appears since each k-rich conic is being counted at least \u0000k 2 \u0001 times, once for each pair of distinct points on the conic. We then bound this by \u0000k 2 \u0001 \u226bk2 (this is certainly valid in the range k \u22655). Using (2.2), we have |Ck| \u226a|P|23/4 k27/4 + |P|4 k4 . (2.3) We proceed to use this estimate to obtain the required incidence bound. We use C=k to denote {C \u2208C: |C \u2229P| = k}. I(P, C) = X k\u22651 |C=k|k = X k\u2264\u2206 |C=k|k + X k>\u2206 |C=k|k \u226a\u2206|C| + X i\u22650 X C\u2208C 2i\u2206\u2264|C\u2229P|<2i+1\u2206 (2i\u2206) \u226a\u2206|C| + X i\u22650 |C2i\u2206|(2i\u2206) \u226a\u2206|C| + X i\u22650   |P|23/4 (2i\u2206)27/4 + |P|4 (2i\u2206)4 ! (2i\u2206) \u226a\u2206|C| + |P|23/4 \u220623/4 + |P|4 \u22063 . In order to optimise the \ufb01rst two terms, we make the choice \u2206= max ( 5, |P|23/27 |C|4/27 ) . This maximum is taken to ensure the application of (2.3) was valid. If the maximum above is 5, then we must have |P|23/27 |C|4/27 \u22645 = \u21d2|P|23 \u226a|C|4. The above bound then becomes I(P, C) \u226a|C| + |P|23/4 \u226a|C|. If the second term in the maximum is taken, we then have I(P, C) \u226a|P|23/27|C|23/27 + |P|13/9|C|12/27. Summing these two bounds to account for either case then gives the result. 6 \f3 Proof of the point-conic bound for Cartesian product sets (Theorem 1.2) The proof of Theorem 1.2 is similar to that of Theorem 1.1. First, we recall an incidence result of Stevens and de Zeeuw. The following version appears in [18, Theorem 5]. We state the result, more generally, over the two dimensional projective space over Fp, denoted by P2(Fp). Theorem 3.1. Given A, B \u2282Fp with |A| \u2264|B|, let P = {[a : b : 1] : (a, b) \u2208A \u00d7 B} \u2282P2(Fp) and let L be a set of lines over P2(Fp). Suppose |A||L| \u226ap2. Then I(P, L) \u226a|A|3/4|B|1/2|L|3/4 + |L| + |A||B|. Corollary 3.2. Let the sets P, L be as in Theorem 3.1 and let \u03c0 be a projective transformation of P2(Fp). Then I(\u03c0(P), L) \u226a|A|3/4|B|1/2|L|3/4 + |L| + |A||B|. Proof. First note that I(\u03c0(P), L) = I(P, \u03c0\u22121(L)). Then the result follows from Lemma 3.1 noting that |L| = |\u03c0\u22121(L)|. Corollary 3.3. For A, B \u2282Fp, with |A| \u2264|B|, let P = {[a : b : 1] : (a, b) \u2208A \u00d7 B} \u2282P2(Fp) and let L denote a set of lines over P2(Fp). Suppose \u03c0 is a projective transformation and for k \u22652, let Lk denote the set of k-rich lines of L with respect to \u03c0(P). Suppose |A||L| \u226ap2. Then |Lk| \u226a|A|3|B|2 k4 + |A||B| k . Proof. The result follows from Corollary 3.2, using the observation that k|Lk| \u226aI(\u03c0(P), Lk). By following the arguments of [32], one obtains the following result on the number of k-rich M\u00a8 obius transformations. Corollary 3.4. For A, B \u2282Fp, with |A| \u2264|B|, let P = A \u00d7 B and let T denote a set of M\u00a8 obius transformations. Suppose \u03c0 is a projective transformation and for k \u22653, let Tk be a the set of k-rich transformations of T, with respect to \u03c0(P). If |A||T| \u226ap2, then |Tk| \u226a|A|4|B|3 k5 + |A|2|B|2 k2 . Proof of Theorem 1.2. Corollary 3.4 may be used, in a similar manner as in the proof of Theorem 1.1, to bound the number of k-rich conics Ck, for k \u22655. This gives |Ck| \u226a|A|6|B|5 k7 + |A|4|B|4 k4 . This can then be converted into the required incidence bound in a similar fashion to the proof of Theorem 1.1. 4 Proof of Theorems 1.3 and 1.4 We begin this section by giving the proof of the circles part of Theorem 1.3, and then explain the alterations necessary to deal with parabolas and hyperbolas. 7 \fProof of Theorem 1.3. Fix a point q \u2208P. We aim to bound the number of k-rich circles passing through q, for k \u22653. After translating the points and circles, we may assume q = (0, 0) without altering the incidences. We rename the translated sets of points and circles as P and C respectively. A circle of the form (x \u2212c)2 + (y \u2212d)2 = r for some c, d, r before this translation, is now of the form Ca,b : (x \u2212a)2 + (y \u2212b)2 = a2 + b2 for some a, b \u2208Fp; this is due to the fact that (0, 0) must lie on the translated circle. Let us take a k-rich circle C. Then there exist points (\u03b11, \u03b21),...,(\u03b1k\u22121, \u03b2k\u22121) \u2208P \\ {(0, 0)} which all lie on C. Each such point can be associated with a line of the form \u22122\u03b1iX \u22122\u03b2iY + \u03b12 i + \u03b22 i = 0. We show that these lines are not de\ufb01ned with multiplicity. Suppose that the two points (\u03b1, \u03b2) and (\u03b1\u2032, \u03b2\u2032) de\ufb01ne the same line. Without loss of generality, we may assume that assume \u03b2 \u0338= 0, since at least one of \u03b1 or \u03b2 is non-zero. After rearranging, we come to the equation Y = \u2212\u03b1 \u03b2 X + \u03b12 + \u03b22 2\u03b2 . We must therefore have \u03b1 \u03b2 = \u03b1\u2032 \u03b2\u2032 , \u03b12 + \u03b22 2\u03b2 = \u03b1\u20322 + \u03b2\u20322 2\u03b2\u2032 setting \u03bb := \u03b1 \u03b2 = \u03b1\u2032 \u03b2\u2032 , we must have \u03b1 = \u03bb\u03b2 and \u03b1\u2032 = \u03bb\u03b2\u2032. Substituting this into the second equation then gives \u03b2(\u03bb2 + 1) = \u03b2\u2032(\u03bb2 + 1). Recalling our assumption p \u22613 (mod 4), it follows that \u22121 is a non-square, and so we conclude that \u03b2 = \u03b2\u2032, which implies \u03b1 = \u03b1\u2032, and so we are done. As we have just seen, the point set P gives rise to a set of lines L, each of the form above, and |P| = |L|. Furthermore, a circle Ca,b as de\ufb01ned above can be associated with the point (a, b). The set of circles C therefore gives rise to a set of points Q \u2286F2 p. Finally, the k-rich circle Ca,b above, corresponds to a (k\u22121)-rich point (a, b) \u2208Q, with respect to the set of lines L. We can see these as the lines corresponding to the points (\u03b1i, \u03b2i) all passing the point (a, b). Now, we use the following corollary of the Stevens\u2013de Zeeuw incidence bound [25, Theorem 3] to bound the number of such (k \u22121)-rich points. Note that this corollary is the dual of [32, Corollary 5]. Corollary 4.1. Let L be a set of lines over F2 p, with |L| \u226ap15/13, and for k \u22652 let Pk denote the number of k-rich points with respect to L. Then |Pk| \u226a|L|11/4 k15/4 + |L| k . Corollary 4.1 and the above argument immediately yields the bound |Cq,k| \u226a|L|11/4 k15/4 + |L| k = |P|11/4 k15/4 + |P| k 8 \fwhere Cq,k denotes the set of k-rich circles, for k \u22653, containing q. Then summing the contribution over all points of P and noting that each k-rich circle gets overcounted by at least a factor of k \u22121 in this way, we have |Ck| \u226a1 k X q\u2208P |Cq,k| \u226a|P|15/4 k19/4 + |P|2 k2 . This can then be used to bound I(P, C) in a similar manner as in the proofs of Theorem 1.1 or [32, Theorem 2]. For parabolas, a similar argument works. After \ufb01xing a point q and translating so that q = (0, 0), we \ufb01nd parabolas of the form y = ax2 + bx. A k-rich parabola then yields a (k \u22121)-rich point (a, b), with respect to the lines given by \u03b2 = X\u03b12 + Y \u03b1 which are again de\ufb01ned with no multiplicity, and the rest of the argument follows similarly to above. Hyperbolas de\ufb01ned by the equation (x \u2212a)(y \u2212b) = c, passing through the point q = (q1, q2), can be written as xy \u2212b(x \u2212q1) \u2212a(y \u2212q2) = q1q2. By setting x\u2032 = x \u2212q1 and y\u2032 = y \u2212q2, our hyperbolas are represented by x\u2032y\u2032 + x\u2032(q2 \u2212b) + y\u2032(q1 \u2212a) = 0. This equation can be viewed as an incidence between the point (q2 \u2212b, q1 \u2212a) and the line x\u2032 \u00b7 X + y\u2032 \u00b7 Y = \u2212x\u2032y\u2032. Thus, we are now in the same situation as before for circles, and the same argument works. The proof of Theorem 1.4 follows from almost exactly the same argument as that of Theorem 1.3, with the only exception being that the dual form of Theorem 3.1 (using point-line duality) replaces Corollary 4.1 to bound the number of k-rich points corresponding to the k-rich circles of C. This is based on the observation that the line set L, as de\ufb01ned in the proof of Theorem 1.3 (corresponding to the point set P in the statement of Theorem 1.4), now has a Cartesian product structure. 5 Proofs of incidence bounds for large sets (Theorems 1.5 and 1.6) To prove Theorem 1.5, we recall the following point-line incidence bound for large sets in [30]. Theorem 5.1. Let P be a set of points and L be a set of lines in F2 q. Then I(P, L) \u2264|P||L| q + q1/2p |P||L|. Corollary 5.2. Let P be a set of points and Lk be the set of k-rich lines over F2 q. Suppose that 9 \fk > |P|/q, then we have |Lk| \u2264q|P| k2 . Utilising Corollary 5.2 and the arguments of [32, Theorem 2], one obtains the following bound on the number of k-rich M\u00a8 obius transformations, which is an analogue of Theorem 2.1 for large sets. Corollary 5.3. Let P be a set of points and T a set of M\u00a8 obius transformations over F2 q. Let Tk be the set of k-rich transformations in T and suppose that k > max{2, |P|/q}. Then |Tk| \u2264q|P|2 k3 . Proof of Theorem 1.5. The proof follows the same approach as in the proof of Theorem 1.1, essentially only replacing Theorem 2.1 by Corollary 5.3 to bound the number of k-rich transformations. In particular, we are able to show that |Ck| \u226a|P|2 k2 \u00b7 q|P|2 k3 = q|P|4 k5 , whenever k > max{4, |P|/q}. Then, writing I(P, C) \u2264 X k\u2264|P| q |C=k|k + X k>max{4, |P| q } |C=k|k + X k\u22644 |C=k|k, we may bound the second sum similarly to the proof of Theorem 1.1, and the \ufb01rst sum trivially, to obtain I(P, C) \u226a|P||C| q + q1/5|P|4/5|C|4/5 + |C|. To prove Theorem 1.6, we require the following bound on incidences between large sets of points and hyperplanes due to Vinh [30]. Theorem 5.4. Let P be a set of points and H be a set of hyperplanes in Fd q. The number of incidences between P and H satis\ufb01es I(P, H) \u2264|P||H| q + q d\u22121 2 (|P||H|)1/2. Proof of Theorem 1.6. The proof uses the same framework as the proof of Theorem 1.3 and so we skip the overly similar details. We begin by \ufb01xing a point q aiming to bound the number of k-rich spheres passing through it. After a translation, we assume q = 0 and so each sphere passing through q takes the form Sa : (x1 \u2212a1)2 + (x2 \u2212a2)2 + \u00b7 \u00b7 \u00b7 + (xd \u2212ad)2 = a2 1 + a2 2 + \u00b7 \u00b7 \u00b7 + a2 d, for some a = (a1, a2, . . . , ad) \u2208Fd q. Let q1, . . . , qk\u22121 denote the k \u22121 points on Sa other than 0 and write qi = (\u03b1(i,1), \u03b1(i,2), . . . , \u03b1(i,d)). For each 1 \u2264i \u2264k \u22121, the point qi can be associated with the hyperplane \u22122\u03b1(i,1)X1 \u22122\u03b1(i,2)X2 \u2212\u00b7 \u00b7 \u00b7 \u22122\u03b1(i,d)Xd + \u03b12 (i,1) + \u03b12 (i,2) + \u00b7 \u00b7 \u00b7 + \u03b12 (i,d) = 0. 10 \fArguing similarly as in the proof of Theorem 1.3, since \u22121 is a non-square, these hyperplanes are de\ufb01ned without multiplicity. Now, having established the correspondence between our original sets of points and spheres to new sets of hyperplanes and points respectively, in order to bound the number, |Sq,k|, of k-rich spheres through q, we require a bound on the number, |Pk|, of k-rich points in terms of hyperplanes over Fd q. To this end, we use Theorem 5.4, to obtain |Pk| \u2264qd\u22121|H| k2 if k > |H|/q. As in the proof of Theorem 1.3, we use the above estimate to bound the number of k-rich spheres, obtaining |Sk| \u2264qd\u22121|P|2 k3 if k > |P|/q. This can then be easily converted into the required incidence bound. 6 Applications 6.1 Pinned algebraic distances Our \ufb01rst application is to the pinned algebraic distance problem. Theorem 6.1. Let f(x, y) be one of the following polynomials: x2 + y2 (usual distance function), xy (Minkowski distance function) or y + x2 (parabola distance function). For E \u2282F2 p with |E| \u226ap15/13 and p \u22613 (mod 4), there exists a point p \u2208E such that |f(p \u2212E)| \u226b|E| 8 15 , where f(p \u2212E) := {f(p \u2212e): e \u2208E}. Remark 6.2. When f(x, y) = x2 + y2, Theorem 6.1 was \ufb01rst proved by Stevens and de Zeeuw in [25] by using a point-line incidence bound. The exponent 8 15 was improved to 1 2 + 149 4214 by Iosevich, Koh, Pham, Shen and Vinh [12], then to 1 2 + 3 74 by Lund and Petridis [16] and to 1 2 + 69 1558 by Iosevich, Koh and Pham [11]. The best current lower bound is |E|2/3 due to Murphy, Petridis, Pham, Rudnev and Stevens [19]. We also note that it seems very di\ufb03cult to extend the methods in [12, 16, 11, 19] to the Minkowski and parabola distance functions. Proof of Theorem 6.1. For p \u2208E, let Cp be the set of conics de\ufb01ned by the equation f(p \u2212x) = t, with t \u2208f(p \u2212E) \\ {0}. Let C = S p\u2208E Cp. We observe that I(E, C) \u226b|E|2. On the other hand, applying Theorem 1.3, we have I(E, C) \u226a(|E||C|)15/19 + |E|23/19|C|4/19 + |C|. Putting the lower and upper bounds of I(E, C) together and using the fact that |C| \u2264P p\u2208E |f(p \u2212 E)|, the theorem follows. We now take advantage of the generality of Theorem 1.1 to study algebraic distances between two sets in F3 p, where one set lies on a plane and the other set is arbitrary. More precisely, let f \u2208Fp[x, y, z] and E, F \u2282F3 p, with E lying on the plane z = 0. As above, the 11 \fset of f-algebraic distances between E and F is de\ufb01ned by f(E \u2212F) := {f(x \u2212y): x \u2208E, y \u2208F}. Theorem 6.3. Let E, F \u2282F3 p, with E lying on the plane z = 0. Suppose f \u2208Fp[x, y, z] satis\ufb01es the property that, for each p \u2208F, the polynomial f(x \u2212p) \u2212t is of degree two and irreducible for all t \u2208f(E \u2212p). We have |f(E \u2212F)| \u226bmin ( |E|4/23|F|4/23, |F|5/4 |E| , |E|20/7 |F| , |E| ) . The following is an example of this theorem. Corollary 6.4. Let f(x, y, z) = x2y2 + z2. For a set E on the plane z = 0 and a set F \u2282F3 p with |F| \u2265|E| 405 216 and p \u22613 (mod 4), we have |f(E \u2212F)| \u226bmin ( |E|4/23|F|4/23, |F|5/4 |E| , |E|20/7 |F| , |E| ) . Remark 6.5. We note that similar questions for some speci\ufb01c polynomials f have been considered in the literature. For instance, the set of distances between a set on a line and an arbitrary set in F2 p was studied by Iosevich, Koh, Pham, Shen and Vinh in [12] as the key step in their improvement of Stevens\u2013de Zeeuw\u2019s result on the original distance problem in two dimensions. Proof of Theorem 6.3. If |f(E \u2212F)| > |E|20/7 |F| , then we are done. Thus, assuming otherwise, we have |f(E \u2212F)| \u00b7 |F| \u2264|E|20/7. For each p \u2208F, let Cp be the set of irreducible conics de\ufb01ned by f(x \u2212p) = t where t \u2208f(E \u2212p). Let C = S p Cp. We observe that I(E, C) \u226b|E||F|. On the other hand, applying Theorem 1.1 implies |E||F| \u226a|E| 23 27 \uf8eb \uf8edX p\u2208F |f(E \u2212p)| \uf8f6 \uf8f8 23/27 + |E|13/9 \uf8eb \uf8edX p\u2208F |f(E \u2212p)| \uf8f6 \uf8f8 12/27 + X p\u2208F |f(E \u2212p)|. Rearranging this inequality gives |f(E \u2212F)| \u226bmin ( |E|4/23|F|4/23, |F|5/4 |E| , |E| ) . We note that the assumption |F| \u2265|E| 405 216 is needed to ensure that |E|19/8 \u2264|C| \u2264|E|20/7 and so the bound on I(E, C), given by Theorem 1.1, is better than the trivial bound (1.4). 6.2 Polynomial images Theorem 6.6. Let f(x, y) be either x2 +y2 (usual distance function) or y +x2 (parabola distance function). In the former case, assume p \u22613 (mod 4). For E, F \u2282F2 p with |E + F| \u226ap15/13, we 12 \fhave |f(E)| \u226bmin ( |E|19/15|F|4/15 |E + F| , |E|19/4|F|15/4 |E + F|23/4 , |E| ) . Proof. We consider the following equation f(x \u2212y) = t, where x \u2208E + F, y \u2208F and t \u2208f(E). Let C be the set of curves de\ufb01ned by f(x \u2212q) = t with q \u2208F and t \u2208f(E). It is not hard to see that if f(x, y) = x2 + y2 or f(x, y) = y + x2, then the curves in C are irreducible. Note that |E||F| \u2264I(E + F, C). Since |C| = |F||f(E)|, using the incidence bound of Theorem 1.3, one has |E||F| \u226a|E + F|15/19(|F||f(E)|)15/19 + |E + F|23/19(|F||f(E)|)4/19 + |F||f(E)|. Solving this inequality completes the proof. Theorem 6.7. Let f(x, y) = xy (Minkowski distance function). For E, F \u2282F2 p with |E + F| \u226a p15/13. Assuming that the two lines x = 0 and y = 0 contain at most |E|/2 points from E, we have |f(E)| \u226bmin ( |E|19/15|F|4/15 |E + F| , |E|19/4|F|15/4 |E + F|23/4 , |E| ) . Proof. Let E\u2032 = E \\ {(a, b) \u2208E : a = 0 or b = 0}. By our assumption, we know that |E\u2032| \u226b|E|. We note that 0 \u0338\u2208f(E\u2032), so the curves de\ufb01ned by (x \u2212a)(y \u2212b) = t, with (a, b) \u2208F and t \u2208f(E\u2032) are irreducible. So, we can use the same argument as in the proof of Theorem 6.6 to conclude the proof of the theorem. Remark 6.8. We note that the assumption that there is at most a proportion of points from E belonging to the two lines x = 0 and y = 0 is necessary, for instance, if E \u2282{x = 0} \u222a{y = 0}, then it is clear that f(E) = {0}. 6.3 An improvement of Koh-Sun\u2019s result on distances for large sets For E, F \u2282Fd q, we denote the set of distances between E and F by the set \u2206(E, F). We recall results of Koh and Sun [15], which remove the logarithmic factor in a result due to Dietmann [7]. In [15, Theorems 3.3], the authors prove that if d \u22653 is odd, then |\u2206(E, F)| \u2265 \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 min n q 2, |E||F| 8qd\u22121 o if 1 \u2264|E| < q d\u22121 2 min \u001a q 2, |F | 8q d\u22121 2 \u001b if q d\u22121 2 \u2264|E| < q d+1 2 min n q 2, |E||F| 2qd o if q d+1 2 \u2264|E| \u2264qd . (6.1) 13 \fFor even d \u22652, under the assumption |E||F| \u226516qd, by [15, Theorems 3.5], one has |\u2206(E, F)| \u2265 \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 q 144 for 1 \u2264|E| < q d\u22121 2 1 144 min \u001a q, |F| 2q d\u22121 2 \u001b for q d\u22121 2 \u2264|E| < q d+1 2 1 144 min n q, 2|E||F| qd o for q d+1 2 \u2264|E| \u2264qd . (6.2) We mention that, in comparison to estimate (6.1) (for odd d), the additional condition |E||F| \u2265 16qd for estimate (6.2) (for even d) is necessary in Koh and Sun\u2019s proof. This is due to the fact that the Fourier decay of the sphere of zero radius in even dimensions is much worse than in odd dimensions. We also note that in the range q d+1 2 \u2264|E| \u2264qd, the lower bound \u226bmin n q, |E||F| qd o was obtained by Shparlinski [24] without the condition |E||F| \u226bqd. As a direct consequence of Theorem 1.6, we are able to remove the condition |E||F| \u226bqd for the range q d\u22121 2 \u2264|E| \u2264q d+1 2 . Theorem 6.9. Let E, F be sets in Fd q. Assume that |E| \u223c|F| \u2264q d+1 2 , then we have |\u2206(E, F)| \u226bmin ( q, |E|1/2|F|1/2 q d\u22121 2 ) . Proof. For p \u2208F, let Cp be the set of spheres centered at p of radius in \u2206(p, E), and let C = S p\u2208F Cp. We have |C| = P p\u2208F |\u2206(p, E)|, and I(E, C) \u226b|E||F|. Therefore, applying Theorem 1.6 gives |E||F| \u226a |E| P p\u2208F |\u2206(p, E)| q + q d\u22121 3 |E|2/3 \uf8eb \uf8edX p\u2208F |\u2206(p, E)| \uf8f6 \uf8f8 2/3 . Solving this inequality, there exists p \u2208F such that |\u2206(p, E)| \u226bmin ( q, |E|1/2|F|1/2 q d\u22121 2 ) . 6.4 Conical Beck\u2019s Theorem A well-known result of Beck [2] states that for a \ufb01nite point set P \u2282R2, either a positive proportion of P is collinear or there exist \u2126(|P|2) distinct lines supported on 2-tuples of (distinct) points of P. See [25, Corollary 14] for a quantitatively weaker analogue of this result, which holds over arbitrary \ufb01elds. We proceed to prove a \ufb01nite \ufb01eld analogue of Beck\u2019s theorem, replacing the notion of lines by irreducible conics. See also [32, Corollary 2] for a similar result involving M\u00a8 obius transformations. In the following, an irreducible conic is said to be de\ufb01ned by a point set P if it passes through at least \ufb01ve points of P. Theorem 6.10. Let P \u2286F2 p be a set of points with |P| \u226ap15/13, with no positive proportion of P collinear. Then either there exists a conic C such that |C \u2229P| \u226b|P|, or P de\ufb01nes at least |P|20/7 irreducible conics. We note that this theorem implies that any point set P \u2286F2 p with p1+\u03b5 < |P| \u226ap15/13, for some 14 \f\u03b5 > 0, must de\ufb01ne at least |P|20/7 irreducible conics, since no conic or line can contain a positive proportion of this point set. We further point out that proof of this theorem relies on a bound on the number of k-rich conics obtained as part of the proof of Theorem 1.1. Moreover, note that a collinearity restriction is necessary in this theorem; an irreducible conic is only de\ufb01ned by \ufb01ve points which lie in general position. Through essentially the same scheme, one may replace this bound by the one obtained in the proof of Theorem 1.3 to give an improved result, concerning circles, parabolas and hyperbolas. Proof of Theorem 6.10. We begin by claiming that a positive proportion of the 5-tuples de\ufb01ned by P are in general position. Indeed, if we let L be the maximum number of collinear points in P, we have #5-tuples in GP = |P|(|P| \u22121)(|P| \u2212L)(|P| \u22123L)(|P| \u22126L) = |P|5 \u2212O(L|P|4) \u226b|P|5, where the last inequality follows from L = o(|P|). From this point we focus only on 5-tuples of points from P which are in general position. With a slight abuse of notation, we write Ck to denote the set of irreducible conics over F2 p containing at least k and at most 2k \u22121 points of P. The number of 5-tuples of points of P contained in elements of Ck is at most O(k5|Ck|) which, by (2.3), is bounded by O(|P|23/4k\u22127/4 + |P|4k). Let I = {k \u22655 : \u03bb|P|3/7 \u2264k \u2264\u03bb\u22127/4|P|} for some constant \u03bb > 0 to be determined. Then, by the above estimate, the total number of 5-tuples of P contained in the conics \u222ak\u2208ICk is at most O(\u03bb\u22127/4|P|5). Taking \u03bb to be su\ufb03ciently large, we may assume these account for less than, say, half of the total number of 5-tuples of P. Moreover, we may assume there exists no irreducible conic containing more than \u03bb\u22127/4|P| points of P, since otherwise there is nothing to prove. We conclude that a positive proportion of the 5-tuples of P lie on conics belonging to the set C := [ k<\u03bb|P|3/7 Ck. Consequently, we have |P|5 \u2248 X C\u2208C |C \u2229P|5 \u226a|C||P|15/7, which gives the second possibility claimed by the theorem, concluding the proof. Acknowledgements Thang Pham would like to thank to the VIASM for the hospitality and for the excellent working condition. Audie Warren was supported by Austrian Science Fund FWF grant P-34180. We thank Oliver Roche-Newton for helpful discussions, and two reviewers for many valuable comments."
+                },
+                {
+                    "url": "http://arxiv.org/abs/2106.07328v2",
+                    "title": "An energy decomposition theorem for matrices and related questions",
+                    "abstract": "Given $A\\subseteq GL_2(\\mathbb{F}_q)$, we prove that there exist disjoint\nsubsets $B, C\\subseteq A$ such that $A = B \\sqcup C$ and their additive and\nmultiplicative energies satisfying \\[ \\max\\{\\,E_{+}(B),\\, E_{\\times}(C)\\,\\}\\ll\n\\frac{|A|^3}{M(|A|)}, \\] where \\begin{equation*} \\label{eqn:MAminBVPolyLSSS}\n  M(|A|) = \\min\\Bigg\\{\\,\\frac{q^{4/3}}{|A|^{1/3}(\\log|A|)^{2/3}},\\,\n\\frac{|A|^{4/5}}{q^{13/5}(\\log|A|)^{27/10}}\\,\\Bigg\\}. \\end{equation*} We also\nstudy some related questions on moderate expanders over matrix rings, namely,\nfor $A, B, C\\subseteq GL_2(\\mathbb{F}_q)$, we have \\[|AB+C|, ~|(A+B)C|\\gg\nq^4,\\] whenever $|A||B||C|\\gg q^{10 + 1/2}$. These improve earlier results due\nto Karabulut, Koh, Pham, Shen, and Vinh (2019).",
+                    "authors": "Ali Mohammadi, Thang Pham, Yiting Wang",
+                    "published": "2021-06-14",
+                    "updated": "2021-06-25",
+                    "primary_cat": "math.CO",
+                    "cats": [
+                        "math.CO",
+                        "math.NT"
+                    ],
+                    "main_content": "Introduction Let Fq denote a \ufb01nite \ufb01eld of order q and characteristic p and let M2(Fq) be the set of two-by-two matrices with entries in Fq. We write X \u226aY or X = O(Y) to mean X \u2264CY for some absolute constant C > 0 and use X \u223cY if Y \u226aX \u226aY. Given subsets A, B \u2286M2(Fq), we de\ufb01ne the sum set A + B to be the set {a + b : (a, b) \u2208A \u00d7 B} and similarly de\ufb01ne the product set AB. In this paper, we study various questions closely related *School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran. Email: a.mohammadi@ipm.ir \u2020Theory of Combinatorial Algorithms Group, ETH Z\u00a8 urich, Switzerland. Email: phamanhthang.vnu@gmail.com \u2021Department of Computer Science, ETH Z\u00a8 urich, Switzerland. Email: yitwang@student.ethz.ch 1 \fto the sum-product problem over M2(Fq), which is to determine nontrivial lower bounds on the quantity max{ |A + A|, |AA| }, under natural conditions on sets A \u2286M2(Fq). A result in this direction was proved by Karabulut et al. in [3, Theorem 1.12], showing that if A \u2286M2(Fq) satis\ufb01es |A| \u226bq3 then max{ |A + A|, |AA| } \u226bmin \u001a |A|2 q7/2 , q2|A|1/2 \u001b . (1) A closely related quantity is the additive energy E+(A, B) de\ufb01ned as the number of solutions (a, a\u2032, b, b\u2032) \u2208A2 \u00d7 B2 such that a + b = a\u2032 + b\u2032. The multiplicative energy E\u00d7(A, B) is de\ufb01ned in a similar manner. We also use, for example, E+(A) = E+(A, A). For \u03bb \u2208M2(Fq), we de\ufb01ne the representation function rAB(\u03bb) = |{ (a, b) \u2208A \u00d7 B : ab = \u03bb }|. Note that rAB is supported on the set AB and so we have the identities \u2211 \u03bb\u2208AB rAB(\u03bb) = |A||B| and \u2211 \u03bb\u2208AB rAB(\u03bb)2 = E\u00d7(A, B). (2) A standard application of the Cauchy-Schwarz inequality gives E+(A, B)|A + B|, E\u00d7(A, B)|AB| \u2265|A|2|B|2. (3) Balog and Wooley [1] initiated the investigation into a type of energy variant of the sum-product problem by proving that given a \ufb01nite set A \u2282R, one may write A = B \u2294C such that max{E+(B), E\u00d7(C)} \u226a|A|3\u2212\u03b4 for some explicit value 0 < \u03b4 < 1. The authors also proved a \ufb01nite \ufb01eld variant of this result. Also, see [2, 4, 9, 10] for quantitative improvements, analogues and various applications of these results. The main goal of this paper is to study energy variants of the sum-product problem over the ring M2(Fq) and in particular to obtain a low energy decomposition result similar to those in [1]. It is important to note that M2(Fq) is not a commutative ring and, in part due to this fact, results known over \ufb01nite \ufb01elds do not readily extend to this setting. 2 Main results Our \ufb01rst theorem is on an energy decomposition of a set of matrices in M2(Fq). 2 \fTheorem 2.1. Given A \u2286GL2(Fq), there exist disjoint subsets B, C \u2286A such that A = B \u2294C and max{E+(B), E\u00d7(C)} \u226a |A|3 M(|A|), where M(|A|) = min ( q4/3 |A|1/3(log |A|)2/3 , |A|4/5 q13/5(log |A|)27/10 ) . (4) It follows from this theorem that for any set A of matrices in M2(Fq), we always can \ufb01nd a subset with either small additive energy or small multiplicative energy. In the setting of \ufb01nite \ufb01elds, such a result has many applications in studying exponential sums and other topics, for instance, see [4, 7, 9, 12, 13, 14, 15] and references therein. By the Cauchy-Schwarz inequality, we have the following direct consequence on a sum-product estimate, namely, for A \u2286GL2(Fq), we have max {|A + A|, |AA|} \u226b|A| \u00b7 M(|A|). (5) By a direct computation, one can check that this is better than the estimate (1) in the range |A| \u226aq3+5/8/(log |A|)1/2. In the next theorem, we show that the lower bound of (5) can be improved by a direct energy estimate. Theorem 2.2. Let A, B \u2286M2(Fq) and C \u2286GL2(Fq). Then E+(A, B) \u226a|A|2|BC|2 q4 + q13/2 |A||BC| |C| . Corollary 2.3. For A \u2286M2(Fq), with |A| \u226bq3, we have max{ |A + A|, |AA| } \u226bmin \u001a |A|2 q13/4 , q4/3|A|2/3 \u001b . (6) In addition, if |AA| \u226a|A| and |A| \u226bq3+ 1 2 , then |A + A| \u226bq4. (7) We point out that the estimate (6) improves (1) in the range |A| \u226aq3+5/8 and is strictly stronger than (5). We also note that our assumption to get the estimate (7) is reasonable. For instance, let G be a subgroup of F\u2217 q, and A be the set of matrices with determinants in G, then we have |A| \u223cq3 \u00b7 |G| and |AA| = |A|. 3 \fIt has been proved in [3, Theorems 1.8 and 1.9] that for A, B, C \u2286M2(Fq), if |A||B||C| \u2265q11, then we have |AB + C|, |(A + B)C| \u226bq4. In the following theorem, we provide improvements of these results. Theorem 2.4. Let A, B, C \u2286M2(Fq), we have |AB + C| \u226bmin \u001a q4, |A||B||C| q13/2 \u001b . If C \u2286GL2(Fq), the same conclusion holds for (A + B)C, i.e. |(A + B)C| \u226bmin \u001a q4, |A||B||C| q13/2 \u001b . In particular, 1. if |A||B||C| \u226bq10+1/2, then |AB + C| \u226bq4. 2. if |A||B||C| \u226bq10+1/2 and C \u2286GL2(Fq), then |(A + B)C| \u226bq4. The condition C \u2286GL2(Fq) is necessary, since, for instance, one can take C to be the set of matrices with zero determinant and A = B = M2(Fq), then |(A + B)C| \u223cq3 and |A||B||C| \u223cq11. We expect that the exponent q10+1/2, in the \ufb01nal conclusions of the above theorem, could be further improved to q10, which, as we shall demonstrate, is sharp. For AB + C, let A and B be the set of lower triangular matrices in M2(Fq) and for arbitrary 0 < \u03b4 < 1, let X \u2286Fq be any set with |X| = q1\u2212\u03b4 and let C = \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8edc1 c2 c3 c4 \uf8f6 \uf8f8: c1, c3, c4 \u2208Fq, c2 \u2208X \uf8fc \uf8fd \uf8fe. Then |A||B||C| = q10\u2212\u03b4 and |AB + C| = |C| = q4\u2212\u03b4. For (A + B)C, the construction is as follows: For arbitrary k, let q = pk and let V denote a (k \u22121)-dimensional vector space over Fp in Fq. Thus, we have |V| = pk\u22121 = q1\u22121/k. Now let A = B = \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8edx1 x2 x3 x4 \uf8f6 \uf8f8: x1, x2 \u2208V, x3, x4 \u2208Fq \uf8fc \uf8fd \uf8fe and C = \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8edc1 c2 c3 c4 \uf8f6 \uf8f8: c1, c3 \u2208Fq, c2, c4 \u2208Fp \uf8fc \uf8fd \uf8fe. 4 \fNote that (A + B)C = AC = \uf8f1 \uf8f2 \uf8f3 \uf8eb \uf8edy1 y2 y3 y4 \uf8f6 \uf8f8: y1, y3, y4 \u2208Fq, y2 \u2208V \uf8fc \uf8fd \uf8fe, where we have used that V \u00b7 Fp + V \u00b7 Fp = V + V = V. Thus, |A||B||C| = (q2 \u00b7 q2\u22122/k)2 \u00b7 (q2 \u00b7 q2/k) = q10\u22122/k while |(A + B)C| = q4\u22121/k. It is worth noting that in the setting of \ufb01nite \ufb01elds, our approach and that of Karabulut et al. in [3] imply the same result. Namely, for A, B, C \u2286Fq, we have |(A + B)C|, |AB + C| \u226bq whenever |A||B||C| \u226bq2. Thus, there exists a different phenomenon between problems over \ufb01nite \ufb01elds and over the matrix ring M2(Fq). Let A, B, C, D \u2286M2(Fq), our last theorem is devoted to the solvability of the equation x + y = zt, x \u2208A, y \u2208B, z \u2208C, t \u2208D. Let J denote the number of solutions to this equation. One can check that by using Theorem 4.2 and Lemma 4.1 from [3], one has \f \f \f \fJ \u2212|A||B||C||D| q4 \f \f \f \f \u226aq7/2(|A||B||C||D|)1/2. (8) Thus, when |A||B||C||D| \u226bq15, then J \u223c|A||B||C||D| q4 . We refer the interested reader to [11] for such a result over \ufb01nite \ufb01elds. In our last theorem, we are interested in bounding J from above when |A||B||C||D| is smaller. Theorem 2.5. Let A, B, C, D \u2286M2(Fq) and let J denote the number of solutions to the equation a + b = cd, (a, b, c, d) \u2208A \u00d7 B \u00d7 C \u00d7 D. Then, we have J \u226a|A||B|1/2|C||D| q2 + q13/4(|A||B||C||D|)1/2. Assuming |A| = |B| = |C| = |D|, the upper bound of this theorem is stronger than that of (8) when |A| \u226aq11/3. Organization. The rest of this paper is structured as follows: In the next section, we prove a preliminary lemma, which is one of the key ingredients in the proof of our energy decomposition theorem. Section 4 is devoted to the proof of Theorem 2.1. The proofs of Theorem 2.2 and Corollary 2.3 will be presented in Section 5. Section 6 contains proofs of Theorem 2.4, and 5 \fTheorem 2.5. 3 A preliminary lemma Given sets A, B, C, D, E, F \u2286M2(Fq), let I(A, B, C, D, E, F) be the number of solutions (a, b, c, d, e, f) \u2208A \u00d7 B \u00d7 C \u00d7 D \u00d7 E \u00d7 F : ab + e f = c + d. The main purpose of this section is to prove an estimate for I(A, B, C, D, E, F), which is one of the key ingredients in the proof of Theorem 2.1. Proposition 3.1. We have \f \f \f \fI(A, B, C, D, E, F) \u2212|A||B||C||D||E||F| q4 \f \f \f \f \u226aq13/2 q |A||B||C||D||E||F| . To prove Proposition 3.1, we de\ufb01ne the sum-product digraph G = (V, E) with the vertex set V = M2(Fq) \u00d7 M2(Fq) \u00d7 M2(Fq), such that there is a directed edge going from (a, e, c) to (b, f, d) if and only if ab + e f = c + d. The setting of this digraph is a generalization of that in [3, Section 4.1] Let G be a digraph on n vertices. Suppose that G is regular of degree d, i.e. the in-degree and out-degree of each vertex are equal to d. Let mG be the adjacency matrix of G, where Mij = 1 if and only if there is a directed edge from i to j. Let \u00b51 = d, \u00b52, . . . , \u00b5n be the eigenvalues of mG. Notice that these eigenvalues can be complex numbers, and for all 2 \u2264i \u2264n, we have |\u00b5i| \u2264d. De\ufb01ne \u00b5(G) := max|\u00b5i|\u0338=d |\u00b5i|. This value is referred to as the second largest eigenvalue of mG. A digraph G is called an (n, d, \u00b5)-digraph if G is a d-regular graph of n vertices, and the second largest eigenvalue is at most \u00b5. We recall the following lemma from [16] on the distribution of edges between two vertex sets on an (n, d, \u00b5)-digraph. Lemma 3.2. Let G = (V, E) be an (n, d, \u00b5)-digraph. For any two sets B, C \u2286V, the number of directed edges from B to C, denoted by e(B, C) satis\ufb01es \f \f \f \fe(B, C) \u2212d n|B||C| \f \f \f \f \u2264\u00b5 q |B||C| . With Lemma 3.2 in hand, to prove Proposition 3.1, it is enough to study properties of the sum6 \fproduct digraph G. De\ufb01nition 3.3. Let a, b \u2208M2(Fq). We say they are equivalent, if whenever the i-th row of a is not all-zero, then neither is the i-th row of b and vice versa, for 1 \u2264i \u22642. Proposition 3.4. The sum product graph G is a (q12, q8, c \u00b7 q13/2)-digraph, for some positive constant c. Proof. The number of vertices is |M2(Fq)|3 = q12. Moreover, for each choice of (b, f), d is determined uniquely from d = ab + e f \u2212c. Thus, there are |M2(Fq)|2 = q8 directed edges going out of each vertex. The number of incoming directed edges can be argued in the same way. To conclude, the digraph G is q8-regular. Let mG denotes the adjacency matrix of G. It remains to bound the magnitude of the second largest eigenvalue of the adjacency matrix of G, i.e., \u00b5(mG). In the next step, we are going to show that mG is a normal matrix, i.e. mT GmG = mGmT G, where mT G is the transpose of mG. For a normal matrix m, we know that if \u03bb is an eigenvalue of m, then |\u03bb|2 is an eigenvalue of mmT and mTm. This comes from the fact that \u03bb is an eigenvalue of mT. Thus, for a normal matrix m, it is enough to give an upper bound for the second largest eigenvalue of mmT or mTm. There is a simple way to check whenever G is normal or not. For any two vertices u and v, let N +(u, v) be the set of vertices w such that \u2212 \u2192 uw, \u2212 \u2192 vw are directed edges, and N \u2212(u, v) be the set of vertices w\u2032 such that \u2212 \u2192 w\u2032u, \u2212 \u2192 w\u2032v are directed edges. It is not hard to check that mG is normal if and only if |N +(u, v)| = |N \u2212(u, v)| for any two vertices u and v. Given two vertices (a, e, c) and (a\u2032, e\u2032, c\u2032), where (a, e, c) \u0338= (a\u2032, e\u2032, c\u2032), the number of (x, y, z) that lies in the common outgoing neighborhood of both vertices is characterized by ax + ey = c + z a\u2032x + e\u2032y = c\u2032 + z \uf8fc \uf8f4 \uf8fd \uf8f4 \uf8fe \u21d0 \u21d2(a \u2212a\u2032)x + (e \u2212e\u2032)y = (c \u2212c\u2032) . For each pair (x, y) satisfying this equation, z is determined uniquely. Thus, the problem is reduced to computing the number of such pairs (x, y). For convenience, let \u00af a = a \u2212a\u2032, \u00af c = c \u2212c\u2032 and \u00af e = e \u2212e\u2032. Also, let t = \u0010 \u00af a \u00af e \u0011 2\u00d74. Then, the above relation is equivalent to \u0010 \u00af a \u00af e \u0011 \uf8eb \uf8edx y \uf8f6 \uf8f8= t \uf8eb \uf8edx y \uf8f6 \uf8f8 4\u00d72 = \u00af c . (9) We now have the following cases: \u2022 (Case 1: rank(t) = 0) Note that in this case, we need a = a\u2032, c = c\u2032 and e = e\u2032, which 7 \fcontradicts our assumption that (a, e, c) \u0338= (a\u2032, e\u2032, c\u2032). Thus, we simply exclude this case. \u2022 (Case 2: rank(t) = 1) As t is not an all-zero matrix, there is at least one non-zero row. Without loss of generality, assume it is the \ufb01rst row. Then, t = \uf8eb \uf8eda1 a2 e1 e2 \u03b1a1 \u03b1a2 \u03b1e1 \u03b1e2 \uf8f6 \uf8f8, where (a1, a2, e1, e2) \u0338= 0 and \u03b1 \u2208Fq. \u2013 (Case 2.1: rank(\u00af c) = 2) In this case, there is no solution, as rank \uf8eb \uf8edt \uf8eb \uf8edx y \uf8f6 \uf8f8 \uf8f6 \uf8f8\u2264 rank(t) = 1 but rank(\u00af c) = 2. \u2013 (Case 2.2: rank(\u00af c) = 1) Let x = \uf8eb \uf8edx1 x2 x3 x4 \uf8f6 \uf8f8, y = \uf8eb \uf8edy1 y2 y3 y4 \uf8f6 \uf8f8. We discuss two sub-cases: (a) \u00af c = \uf8eb \uf8edc1 c2 \u03b1c1 \u03b1c2 \uf8f6 \uf8f8with the same factor \u03b1, where (c1, c2) \u0338= (0, 0). In this case, we have the following set of equations: \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 a1x1 + a2x3 + e1y1 + e2y3 = c1 a1x2 + a2x4 + e1y2 + e2y4 = c2 . Since we assume (a1, a2, e1, e2) \u0338= 0, without loss of generality, let a1 \u0338= 0. Then, \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 x1 = (a1)\u22121(c1 \u2212a2x3 \u2212e1y1 \u2212e2y3) x2 = (a1)\u22121(c2 \u2212a2x4 \u2212e1y2 \u2212e2y4) , which means that for each (x3, y1, y3) there is a unique x1 and for each (x4, y2, y4) there is a unique x2. Thus, there are q6 different (x, y, z) solutions. (b) In all other sub-cases, there is no solution. If \u00af c = \uf8eb \uf8edc1 c2 \u03b2c1 \u03b2c2 \uf8f6 \uf8f8, where \u03b2 \u0338= \u03b1 and (c1, c2) \u0338= (0, 0), then we get the following two equations: \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 a1x1 + a2x3 + e1y1 + e2y3 = c1 \u03b1a1x1 + \u03b1a2x3 + \u03b1e1y1 + \u03b1e2y4 = \u03b2c1 , which obviously do not have any solution. Otherwise, \u00af c = \uf8eb \uf8ed\u03b2c1 \u03b2c2 c1 c2 \uf8f6 \uf8f8where (c1, c2) \u0338= (0, 0). Note that if \u03b1 \u0338= 0, then \u03b2 \u0338= \u03b1\u22121, 8 \fbecause this case is covered in Case 2.2(a) implicitly. We get the following equations. \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 a1x1 + a2x3 + e1y1 + e2y3 = \u03b2c1 \u03b1a1x1 + \u03b1a2x3 + \u03b1e1y1 + \u03b1e2y3 = c1 , which obviously do not have any solution. Notice that \u03b1 = 0 or \u03b2 = 0 corresponds to t and \u00af c not being equivalent. \u2013 (Case 2.3: rank(\u00af c) = 0) This case is similar to the Case 2.2(a), except c1 = c2 = 0. We have the following two equations: \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 a1x1 + a2x3 + e1y1 + e2y3 = 0 a1x2 + a2x4 + e1y2 + e2y4 = 0 . Following the same analysis, we conclude there are q6 solutions. \u2022 (Case 3: rank(t) = 2) In this case, we always have solutions, for any \u00af c. \u2013 (Case 3.1: rank(\u00af a) = 2 or rank(\u00af e) = 2) In this case, let us look back on equation (9). If rank(\u00af a) = 2, then we can rewrite (9) as \u00af ax = \u00af c \u2212\u00af ey. Observe that, for any y \u2208M2(Fq), there is a unique x. Thus, the number of solutions is q4. The case where rank(\u00af e) = 2 is similar. \u2013 (Case 3.2: rank(\u00af a) \u22641 and rank(\u00af e) \u22641) In this case, it is not hard to observe that T must be one of the following four types: (i) \uf8eb \uf8eda1 a2 e1 e2 \u03b1a1 \u03b1a2 \u03b2e1 \u03b2e2 \uf8f6 \uf8f8, where (a1, a2), (e1, e2) \u0338= (0, 0), \u03b1 \u0338= \u03b2, (\u03b1, \u03b2) \u0338= (0, 0). (ii) \uf8eb \uf8ed\u03b1a1 \u03b1a2 \u03b2e1 \u03b2e2 a1 a2 e1 e2 \uf8f6 \uf8f8, where (a1, a2), (e1, e2) \u0338= (0, 0), \u03b1 \u0338= \u03b2, (\u03b1, \u03b2) \u0338= (0, 0). (iii) \uf8eb \uf8eda1 a2 0 0 0 0 e1 e2 \uf8f6 \uf8f8, where (a1, a2), (e1, e2) \u0338= (0, 0). (iv) \uf8eb \uf8ed0 0 e1 e2 a1 a2 0 0 \uf8f6 \uf8f8, where (a1, a2), (e1, e2) \u0338= (0, 0). Since (i) and (ii) are symmetric and so is (iii) and (iv), we only argue for (i) and (iii). 9 \fFor (iii), reusing notations from Case 2.2(a), we have \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 a1x1 + a2x3 = c1 a1x2 + a2x4 = c2 e1y1 + e2y3 = c3 e1y2 + e2y4 = c4 . As (a1, a2) \u0338= (0, 0) and (e1, e2) \u0338= (0, 0), without loss of generality, we assume a1 \u0338= 0 and e1 \u0338= 0. Then, it means for each (x3, x4, y3, y4) there is a unique (x1, x2, y1, y2). Thus, the system has q4 solutions. For (i), we have \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 a1x1 + a2x3 + e1y1 + e2y3 = c1 1 a1x2 + a2x4 + e1y2 + e2y4 = c2 2 \u03b1a1x1 + \u03b1a2x3 + \u03b2e1y1 + \u03b2e2y3 = c3 3 \u03b1a1x2 + \u03b1a2x4 + \u03b2e1y2 + \u03b2e2y4 = c4 4 . Again, assume a1 \u0338= 0 and e1 \u0338= 0. Now, take 1 \u00d7 \u03b1 \u22123 , we get (\u03b1 \u2212\u03b2)(e1y1 + e2y3) = \u03b1c1 \u2212c3. As \u03b1 \u0338= \u03b2, this means e1y1 + e2y3 = (\u03b1 \u2212\u03b2)\u22121(\u03b1c1 \u2212c3). Thus, for each y3, there is a unique y1. Similarly, compute 1 \u00d7 \u03b2 \u22123 , and we get a1x1 + a2x3 = (\u03b2 \u2212\u03b1)\u22121(\u03b2c1 \u2212c3), which means that for each x3, we get a unique x1. We can do the same for 2 and 4 and conclude that there are q4 solutions. Observe that all cases are disjoint and they together enumerate all possible relations between vertices (a, e, c) and (a\u2032, e\u2032, c\u2032). We computed N +((a, e, c), (a\u2032, e\u2032, c\u2032)) at above and the computation for N \u2212((a, e, c), (a\u2032, e\u2032, c\u2032)) is similar. Thus, we know mG is normal. Note that each entry of mGmT G can be interpreted as counting the number of common outgoing neighbors between two vertices. We can write mGmT G as mGmT G = (q8 \u2212q4)I + q4J \u2212q4E21 + (q6 \u2212q4)E22a \u2212q4E22b + (q6 \u2212q4)E23 + (q4 \u2212q4)E31 + (q4 \u2212q4)E32 = (q8 \u2212q4)I + q4J \u2212q4E21 + (q6 \u2212q4)E22a \u2212q4E22b + (q6 \u2212q4)E23 , where I is the identity matrix, J is the all one matrix and Eijs are adjacency matrices, specifying which entries are involved. For example, for Case 2.3, all pairs (a, e, c), (a\u2032, e\u2032, c\u2032) with c = c\u2032 and 10 \frank(t) = 1 are involved. Thus, E23 is an adjacency matrix of size q12 \u00d7 q12 (containing all pairs (a, e, c)), with pairs of vertices satisfying this property marked 1 and all others marked 0. Finally, observe that each subgraph de\ufb01ned by the corresponding adjacency matrix Eij is regular. This is due to the fact that the condition does not depend on speci\ufb01c value of (a, e, c). Starting from any vertex (a, e, c), we can get to all possible \u00af a, \u00af e, \u00af c by subtracting the correct (a\u2032, e\u2032, c\u2032). Thus, for each case, we get the same number of (a\u2032, e\u2032, c\u2032) that satis\ufb01es the condition. Let \u03baij denotes an upper bound on the magnitude of the largest eigenvalue of Eij. Then, it is easy to see that \u03ba21 \u226aq9, \u03ba22a \u226aq7, \u03ba22b \u226aq8 and \u03ba23 \u226aq5. For example, in Case 2.1, we have rank(t) = 1 and rank(\u00af c) = 2. For a \ufb01xed (a, e, c), the former implies that there are O(q5) possibilities for a\u2032 and e\u2032 while the latter implies there are O(q4) possibilities for c\u2032. Altogether, there are O(q9) possibilities for (a\u2032, e\u2032, c\u2032) in Case 2.1. Because the graph induced by E21 is regular, we have \u03ba21 \u226aq9. Other cases can be deduced accordingly. The rest follows from a routine computation: let v2 be an eigenvector corresponding to \u00b5(G). Then, because G is regular and connected (easy to see, there is no isolated vertex), v2 is orthogonal to the all 1 vector, which means J \u00b7 v2 = 0. We now have \u00b5(mG)2v2 = mGmT G \u00b7 v2 = (q8 \u2212q4)I \u00b7 v2 + (\u2212q4E21 + (q6 \u2212q4)E22a \u2212q4E22b + (q6 \u2212q4)E23) \u00b7 v2 \u226aq13 \u00b7 v2 . Thus, \u00b5(G) \u226aq13/2. Proof of Proposition 3.1. It follows directly from Proposition 3.4 and Lemma 3.2 that \f \f \f \fI(A, B, C, D, E, F) \u22121 q4 |A||B||C||D||E||F| \f \f \f \f \u226aq13/2 q |A||B||C||D||E||F| . This completes the proof. 4 Proof of Theorem 2.1 To prove Theorem 2.1, we will also need several technical results. A proof of the following inequality may be found in [9, Lemma 2.4]. Lemma 4.1. Let V1, . . . , Vk be subsets of an Abelian group. Then E+ \u0012 k G i=1 Vi \u0013 \u2264 \u0012 k \u2211 i=1 E+(Vi)1/4 \u00134 . 11 \fThe following lemma is taken from [4] and may also be extracted from [9] and [10]. Lemma 4.2 is slightly different to its analogues over commutative rings as highlighted by the duality of the inequalities (14) and (15). Lemma 4.2. Let X \u2286GL2(Fq). There exist sets X\u2217\u2282X, D \u2282XX, as well as numbers \u03c4 and \u03ba satisfying E\u00d7(X) 2|X|2 \u2264\u03c4 \u2264|X|, (10) E\u00d7(X) \u03c42 \u00b7 log |X| \u226a|D| \u226a(log |X|)6 |X\u2217|4 E\u00d7(X), (11) |X\u2217|2 \u226b E\u00d7(X) |X|(log |X|)7/2 , (12) \u03ba \u226b |D|\u03c4 |X\u2217|(log |X|)2 (13) such that either rDX\u22121(x) \u2265\u03ba for all x \u2208X\u2217. (14) or rX\u22121D(x) \u2265\u03ba for all x \u2208X\u2217. (15) We need a dyadic pigeonhole argument, which can be found in [6, Lemma 18]. Lemma 4.3. For \u2126\u2286M2(Fq), let w, f : \u2126\u2192R+ with f(x) \u2264M, \u2200x \u2208\u2126. Let W = \u2211x\u2208\u2126w(x). If \u2211x\u2208\u2126f(x)w(x) \u2265K, then there exists a subset D \u2282\u2126and a number \u03c4 such that \u03c4 \u2264f(x) < 2\u03c4 for all x \u2208D and K/(2W) \u2264\u03c4 \u2264M . Moreover K 2 + 2 log2 M \u2264\u2211 x\u2208D f(x)w(x) \u22642\u03c4 \u2211 x\u2208D w(x) \u2264min{2\u03c4W, 4\u03c42|D|} . Proof of Lemma 4.2. We use the identities in (2) and apply Lemma 4.3, by taking \u2126= XX, f = w = rXX, M = |X|, K = E\u00d7(X) and W = |X|2, to \ufb01nd a set D \u2282XX and a number \u03c4, satisfying (10), such that D = { \u03bb \u2208XX : \u03c4 \u2264rXX(\u03bb) < 2\u03c4 } and \u03c42|D| \u226bE\u00d7(X)/ log |X| . (16) De\ufb01ne P1 = { (x, y) \u2208X \u00d7 X : xy \u2208D } and Ax = { y : (x, y) \u2208P1 } for x \u2208X. By the de\ufb01nition of D, we know that \u03c4|D| \u2264|P1| < 2\u03c4|D|. We can use Lemma 4.3 again with \u2126= X, f(x) = |Ax|, w = 1, M = W = |X| and K = |P1| to \ufb01nd a set V \u2282X and a number \u03ba1 such 12 \fthat V = { x \u2208X : \u03ba1 \u2264|Ax| < 2\u03ba1 } and |V|\u03ba1 \u226b|P1|/ log |X| \u226b\u03c4|D|/ log |X| . (17) Now we split the analysis into two cases based on |V|: Case 1 (|V| \u2265\u03ba1(log |X|)\u22121/2): In this case, we simply set X\u2217= V and \u03ba = \u03ba1. For each x \u2208V, there are at least \u03ba1 different y such that xy \u2208D. Therefore, rDX\u22121(x) \u2265\u03ba \u2200x \u2208X\u2217. Case 2 (|V| < \u03ba1(log |X|)\u22121/2): In this case, we \ufb01nd another pair U, \u03ba2 that satis\ufb01es |U| \u226b \u03ba2(log |X|)\u22121/2 and set X\u2217= U and \u03ba = \u03ba2. Let P2 = { (x, y) \u2208P1 : x \u2208V } and By = { x : (x, y) \u2208P2 }. By de\ufb01nition, we have |P2| \u2265|V|\u03ba1. We apply Lemma 4.3 again, with \u2126= X, f(y) = |By|, w = 1, K = |P2|, W = M = |X| to get U \u2282X and a number \u03ba2 such that U = { y \u2208X : \u03ba2 \u2264|By| < 2\u03ba2 } and |U|\u03ba2 \u226b|P2|/ log |X| \u2265\u03ba1|V|/ log |X| . (18) Combining this inequality with the assumption of this case (\u03ba1 \u2265|V|(log |X|)1/2) and |V| \u2265\u03ba2, we conclude |U| \u226b\u03ba2(log |X|)\u22121/2. We can then argue similarly as in Case 1 to conclude rX\u22121D(x) \u2265\u03ba \u2200x \u2208X\u2217. Now, (13) follows from either of (17) or (18). To prove (12), we \ufb01rst note that in either of the cases above we have |X\u2217| \u226b\u03ba(log |X|)\u22121/2. Then using the lower bound on \u03ba, (16) and (10), we have |X\u2217|2 \u226b|D|\u03c4(log |X|)\u22125/2 \u226bE\u00d7(X)/(|X| log |X|)7/2 as required. Finally, to deduce the required upper bound on |D| in (11) note that, as shown above, |D|\u03c4 \u226a|X\u2217|2(log |X|)5/2, which with (16) implies |D|E\u00d7(X)(log |X|)\u22121 \u226a(|D|\u03c4)2 \u226a|X\u2217|4(log |X|)5. Lemma 4.4. Let X \u2286GL2(Fq). Then there exists X\u2217\u2286X, with |X\u2217| \u226b E\u00d7(X)1/2 |X|1/2(log |X|)7/4 , such that E+(X\u2217) \u226a|X\u2217|4|X|6(log |X|)2 q4E\u00d7(X)2 + q13/2|X\u2217|3|X|(log |X|)5 E\u00d7(X) . (19) Proof. We apply Lemma 4.2 to the set X and henceforth assume its full statement, keeping the 13 \fsame notation. Without loss of generality, assume rX\u22121D \u2265\u03ba \u2200x \u2208X\u2217. Thus, E+(X\u2217) = |{(x1, x2, x3, x4) \u2208X4 \u2217: x1 + x2 = x3 + x4}| \u2264\u03ba\u22122|{(d1, d2, x1, x2, y1, y2) \u2208D2 \u00d7 X2 \u2217\u00d7 X2 : x1 + y\u22121 1 d1 = x2 + y\u22121 2 d2}| = \u03ba\u22122I(X\u22121, D, \u2212X\u2217, \u2212X\u22121, D, X\u2217). Then applying Proposition 3.1 and (13), we obtain E+(X\u2217) \u226a\u03ba\u22122 \u00b7 \u0012(|D||X||X\u2217|)2 q4 + q13/2|D||X||X\u2217| \u0013 \u226a|X\u2217|4|X|2(log |X|)2 q4\u03c42 + q13/2|X\u2217|3|X|(log |X|)4 |D|\u03c42 . Finally, applying (10) and (11), we obtain the required bound in (19) for E+(X\u2217). We are now ready to give a proof of Theorem 2.1. Proof of Theorem 2.1. We begin by describing an algorithm, which constructs two sequences of sets A = S1 \u2287S2 \u00b7 \u00b7 \u00b7 \u2287Sk+1 and \u2205= T0 \u2286T1 \u00b7 \u00b7 \u00b7 \u2286Tk such that Si \u2294Ti\u22121 = A, for i = 1, . . . , k + 1. Let 1 \u2264M \u2264|A| be a parameter. At any step i \u22651, if E\u00d7(Si) \u2264|A|3/M the algorithm halts. Otherwise if E\u00d7(Si) > |A|3 M , (20) through a use of Lemma 4.4, with X = Si, we identify a set Vi := X\u2217\u2286Si, with |Vi| \u226b E\u00d7(Si)1/2 |Si|1/2(log |A|)7/4 > |A| M1/2(log |A|)7/4 (21) and E+(Vi) \u226a|Vi|4|Si|6(log |Si|)2 q4E\u00d7(Si)2 + q13/2|Vi|3|Si|(log |Si|)5 E\u00d7(Si) . (22) We then set Si+1 = Si \\ Vi, Ti+1 = Ti \u2294Vi and repeat this process for the step i + 1. From (21), we deduce |Vi| \u226b|A|1/2(log |A|)\u22127/4 and so the cardinality of each Si monotonically decreases. This in turn implies that this process indeed terminates after a \ufb01nite number of iterations k. We set B = Sk+1 and C = Tk, noting that A = B \u2294C and that E\u00d7(B) \u2264|A|3 M . (23) 14 \fWe apply the inequalities (20), (21) and |Si| \u2264|A|, to (22), to get E+(Vi) \u226aM2|Vi|4q\u22124(log |A|)2 + M|A|\u22122|Vi|3q13/2(log |A|)5 \u226a \u0000M2q\u22124(log |A|)2 + M3/2|A|\u22123q13/2(log |A|)27/4\u0001 \u00b7 |Vi|4. Then, observing that C = Tk = k G i=1 Vi \u2286A, we use Lemma 4.1 to obtain E+(C) \u226a(M2q\u22124(log |A|)2 + M3/2|A|\u22123q13/2(log |A|)27/4) \u0012 k \u2211 i=1 |Vi| \u00134 \u2264M2|A|4q\u22124(log |A|)2 + M3/2|A|q13/2(log |A|)27/4. Note that Lemma 4.1 is applicable because M2(Fq) is an Abelian group under addition. Comparing this with (23), we see the choice M = M(|A|), given by (4) is optimal. 5 Proofs of Theorem 2.2 and Corollary 2.3 Proof of Theorem 2.2. We proceed similarly to the proof of [8, Theorem 6]. Note that E+(A, B) = |C|\u22122|{ (a, a\u2032, b, b\u2032, c, c\u2032) \u2208A2 \u00d7 B2 \u00d7 C2 : a + bcc\u22121 = a\u2032 + b\u2032c\u2032(c\u2032)\u22121 }| \u2264|C|\u22122|{ (a, a\u2032, s, s\u2032, c, c\u2032) \u2208A2 \u00d7 (BC)2 \u00d7 (C\u22121)2 : a + sc = a\u2032 + s\u2032c\u2032 }|. The required result then follows by applying Proposition 3.1. Proof of Corollary 2.3. Since |A| \u226bq3, we may assume A \u2286GL2(Fq). We use Theorem 2.2, with A = B = C and apply the lower bound on E+(A) given by (3) to obtain (6). To prove (7), we follow the same process and apply the assumption |AA| \u226a|A|, to obtain |A + A| \u226b min{ q4, |A|3/q13/2 }, which gives the required result. 6 Proofs of Theorem 2.4 and Theorem 2.5 Proof of Theorem 2.4. For \u03bb \u2208AB + C, write t(\u03bb) = |{ (a, b, c) \u2208A \u00d7 B \u00d7 C : ab + c = \u03bb }|. 15 \fBy the Cauchy-Schwarz inequality, we have (|A||B||C|)2 =   \u2211 \u03bb\u2208AB+C t(\u03bb) !2 \u2264|AB + C| \u2211 \u03bb\u2208AB+C t(\u03bb)2. Further noting that \u2211 \u03bb\u2208AB+C t(\u03bb)2 = I(A, B, \u2212C, \u2212A, B, C). We apply Proposition 3.1 to obtain |AB + C| \u226bmin \u001a q4, |A||B||C| q13/2 \u001b . This immediately implies the required result. For the set (A + B)C, as above we have |(A + B)C| \u2265 |A|2|B|2|C|2 |{ (a, b, c, a\u2032, b\u2032, c\u2032) \u2208(A \u00d7 B \u00d7 C)2 : (a + b)c = (a\u2032 + b\u2032)c\u2032 }|. To estimate the denominator, we follow the argument in the proof of Proposition 3.1. In particular, we \ufb01rst de\ufb01ne a graph G with the vertex set V = M2(Fq) \u00d7 M2(Fq) \u00d7 M2(Fq), and there is a direct edge going from (a, e, c) to (b, f, d) if ba + e f = c + d. The only difference here compared to that graph in Section 3 is that we switch between ba and ab. By using a similar argument as in Section 3, we have this graph is a (q12, q8, cq13/2)-digraph, where c is a positive constant. To bound the denominator, we observe that the equation (a + b)c = (a\u2032 + b\u2032)c\u2032 gives us a direct edge from (c, \u2212b\u2032, \u2212ac) to (b, c\u2032, a\u2032c\u2032). So let U := {(c, \u2212b\u2032, \u2212ac): a \u2208A, c \u2208 C, b\u2032 \u2208B} and W = {(b, c\u2032, a\u2032c\u2032): b \u2208B, c\u2032 \u2208C, a\u2032 \u2208A}. Since C \u2286GL2(Fq), we have |U| = |W| = |A||B||C|. So applying Lemma 3.2, the number of edges from U to W is at most |A|2|B|2|C|2 q4 + q13/2|A||B||C|. In other words, |{ (a, b, c, a\u2032, b\u2032, c\u2032) \u2208(A \u00d7 B \u00d7 C)2 : (a + b)c = (a\u2032 + b\u2032)c\u2032 }| \u226a|A|2|B|2|C|2 q4 + q13/2|A||B||C|, and we get the desired estimate. 16 \fProof of Theorem 2.5. By the Cauchy-Schwarz inequality and Proposition 3.1, we have J = |{ (a, b, c, d) \u2208A \u00d7 B \u00d7 C \u00d7 D : a + b = cd }| \u2264|B|1/2|{ (a, a\u2032, c, c\u2032, d, d\u2032) \u2208A2 \u00d7 C2 \u00d7 D2 : cd \u2212a = c\u2032d\u2032 \u2212a\u2032 }|1/2 \u226a|A||B|1/2|C||D| q2 + q13/4(|A||B||C||D|)1/2. Acknowledgments Thang Pham was supported by Swiss National Science Foundation grant P4P4P2-191067."
+                }
+            ],
+            "Thang Pham": [
+                {
+                    "url": "http://arxiv.org/abs/2308.01504v3",
+                    "title": "New-type Quasirandom Groups and Applications",
+                    "abstract": "This paper aims to introduce a more general definition of quasirandom groups\nand generalize several well-known results in the literature in this new\nsetting. More precisely, let $G$ be a semi-direct product of groups and\n$X\\subseteq G$, we provide conditions such that one can find tuples $(x_0,\n\\ldots, x_k)\\in X^{k+1}$ satisfying $x_1x_2\\ldots x_k=x_0$ or conditions to\nguarantee that the product set $XX$ grows exponentially. In a special case of\nthe group of rigid-motions in the plane over an arbitrary finite field, our\nresults offer a reasonably complete description of structures of this group.",
+                    "authors": "Thang Pham, Boqing Xue",
+                    "published": "2023-08-03",
+                    "updated": "2023-08-25",
+                    "primary_cat": "math.CO",
+                    "cats": [
+                        "math.CO",
+                        "math.NT"
+                    ],
+                    "main_content": "Introduction Let (G, \u00b7) be a \ufb01nite group and D be a positive integer. The group G is called D-quasirandom if all non-trivial representations of G are of degree at least D. This notion was introduced by Gowers [5] in his solution of the following question due to Babai and S\u00b4 os [2]: Does there exist a constant c > 0 such that every \ufb01nite group G has a product-free subset of size at least c|G|? Gowers proved the following theorem. Theorem 1.1. If G is D-quasirandom, then \f \f \f \f#{(x, y, xy) \u2208X \u00d7 Y \u00d7 Z} \u2212|X||Y ||Z| |G| \f \f \f \f \u226a r |G||X||Y ||Z| D (1) for every X, Y, Z \u2286G. This means that when X, Y and Z are large enough, then the number of triples (x, y, xy) \u2208 X \u00d7 Y \u00d7 Z is close to the expected value. As a direct application, if X \u2286G is a product-free subset of G, then we have |X| \u226a|D|\u22121/3|G|. Gowers also asked whether a similar result holds for three-term progressions, i.e. triples of the form (x, xy, xy2). This question was solved by Tao [9] for SLd(Fq), Peluse [8] for non-abelian \ufb01nite simple groups, and by Bhangale, Harsha and Roy [3] for all \ufb01nite quasirandom groups. Another interesting application of the estimate (1) is on the growth of product of sets in quasirandom groups. More precisely, for X, Y \u2286G, we set Z = XY , then |XY | \u226bmin \u001a |G|, D |G||X||Y | \u001b . (2) 1University of Science, Vietnam National University, Hanoi. Email: phamanhthang.vnu@gmail.com 2Institute of Mathematical Sciences, ShanghaiTech University. Email: xuebq@shagnhaitech.edu.cn 1 \fWe note that a more general statement of this result in the form of ||1X \u22171Y ||2 2 was studied by Babai, Nikolov, and Pyber [1]. In the setting of the group SL2(Fq), it is well\u2013known that D = (q \u22121)/2 (see [4], page 102). The estimate (2) gives |XX| \u226bmin \b q3, q\u22122|X|2\t , so any subset X of SL2(Fq) with |X| = q\u03b1 (\u03b1 > 2) has exponentially product growth. It is worth noting that computing the product of elements from a group is a fundamental problem in theoretical computer science. Gowers and Viola [6] studied mixing in several non-quasirandom groups, and obtained results on communication complexity. One of the models considered in their paper is the a\ufb03ne group A\ufb00(Fq) over Fq, which is a semi-product group. Since this group has large subgroups, the estimates (1) and (2) do not hold anymore. The main purpose of this paper is to extend the estimate (1) in the setting of semi-direct product of groups, i.e. groups G of the form N \u22ca\u03d5 H. If H = {1} (the trivial multiplicative group), then our result recovers the estimate (1). Our initial motivation of studying this topic comes from the following question: given a set X of rigid-motions in the plane over an arbitrary \ufb01nite \ufb01eld Fq, under what conditions on X does the set XX grow exponentially? We start with a simple observation. Consider the group G0 = F2 q \u22ca\u03d5 SO2(Fq) of rigid motions with |G0| = q2(q \u00b1 1). (See more details in section 1.2 and Example 1.10.) Let \u03b3 be a generator of the cyclic group SO2(Fq). Assume that |SO2(Fq)| = q \u00b1 1 = kl, and let X = {(z, \u03b3kj) : z \u2208F2 q, 0 \u2264j \u2264l \u22121}. Then X is a subgroup of G0 and XX = X. Here |X|/|G0| = 1/k. This infers that, it is possible to choose an arbitrarily large subset X \u2286G0 such that the product XX does not grow in general. In other words, a condition on the size of X is not enough to guarantee the expanding property of XX. The second observation we want to mention here is that, one can apply the estimate (2) on G0 to obtain that |X| = |XX| \u226bmin \u001a q3, D q3 |X|2 \u001b \u226bD k |X|. The above example suggests that the number D for the group of rigid motions should be at most a constant which does not depend on q. This is true and will be con\ufb01rmed in Theorem 1.7 below. Putting these two observations together, we realize that in order to understand structures of the product set XX, a deeper studying is needed. In this paper, our results will be stated and proved in the setting of semi-direct product groups. The group of rigid-motions is a special case. The main tool we will use is non-abelian Fourier analysis. 2 \f1.1 Main results on semi-direct product groups Let (N, \u00b7) and (H, \u00b7) be two groups with identities 1N and 1H, respectively. Assume that \u03d5 : H \u2192Aut(N) is a group homomorphism. For simplicity, we denote \u03d5h = \u03d5(h) for h \u2208H. The semi-direct product (G, \u00b7) = (N, \u00b7) \u22ca\u03d5 (H, \u00b7) is a group of order |N||H|. More explicitly, one has G = {(z, h) : z \u2208N, h \u2208H} with the group law given by (z1, h1)(z2, h2) = (z1 \u03d5h1(z2), h1h2). Through this paper, for each g \u2208G, we write g := (\u00a8 g, \u02d9 g) or g = (g\u00b7\u00b7, g\u00b7). The subgroup e N := {(z, 1H) : z \u2208N} is a normal subgroup of G, which gives G/ e N \u2243H. Denote by \u03c0 : G \u2192G/ e N the quotient map. If \u03c1 : G \u2192GL(V ) is a complex representation of G such that \u03c1| e N is the identity, then Ker(\u03c1) \u2287e N. As a result, there is a unique homomorphism \u02d9 \u03c1 : G/ e N \u2192GL(V ) such that \u03c1 = \u02d9 \u03c1\u03c0. Here, we can view \u02d9 \u03c1 as a representation of H. On the other hand, for any representation \u02d9 \u03c1 of H \u2243G/ e N, the homomorphism \u03c1 = \u02d9 \u03c1\u03c0 is a representation of G. Now all the representations of G can be divided into the following two types: Type I: a representation \u03c1 that is lifted from some representation \u02d9 \u03c1 of H; Type II: a representation \u03c1 such that \u03c1((z, 1H)) is not the identity for some z \u2208N. We are interested in the situation that type II representations have large degree. De\ufb01nition 1.2. Let D be a positive number. We say that G = N \u22ca\u03d5 H is D-quasirandom on N if the degree of any representation of G of type II is at least D. We note that if a group G\u2032 is D-quasirandom in Gowers\u2019s de\ufb01nition, then it is D-quasirandom in this de\ufb01nition by viewing G\u2032 \u2243G\u2032\u22ca{1}. However, the inverse is not true, the group of rigid-motions (with Theorem 1.7 below) is an example. Our \ufb01rst result states as follows. Theorem 1.3. Let G = N \u22ca\u03d5 H be de\ufb01ned as previous, which is D-quasirandom on N. For any k \u22652 and X0, X1, . . . , Xk \u2286G, denote Mk = #{(x0, x1, . . . , xk) \u2208X0 \u00d7 X1 \u00d7 . . . \u00d7 Xk : x1x2 \u00b7 \u00b7 \u00b7 xk = x0}, and \u02d9 Mk = #{(x0, x1, . . . , xk) \u2208X0 \u00d7 X1 \u00d7 . . . \u00d7 Xk : \u02d9 x1 \u02d9 x2 \u00b7 \u00b7 \u00b7 \u02d9 xk = \u02d9 x0}. 3 \fThen \f \f \f \fMk \u2212 1 |N| \u02d9 Mk \f \f \f \f \u2264 r |G|k\u22121|X0||X1| . . . |Xk| Dk\u22121 . Assume that G\u2032 is D-quasirandom. Taking k = 2 and G = G\u2032 \u22ca{1} in the above theorem, one has \u02d9 M2 = |X0||X1||X2|. Then Theorem 1.1 follows immediately from Theorem 1.3. For any subset X \u2286G, denote by 1X the characteristic function of X, i.e., 1X(x) = \uf8f1 \uf8f2 \uf8f3 1, if x \u2208X, 0, if x / \u2208X. The next theorem shows upper and lower bounds for convolutions. Theorem 1.4. Let G = N \u22ca\u03d5 H be de\ufb01ned as previous, which is D-quasirandom on N. For any integer k \u22652 and subsets X1, X2, . . . , Xk of G, we have |H| \u02d9 Nk |G|2k \u2264\u22251X1 \u22171X2 \u2217. . . \u22171Xk\u22252 2 \u2264|H| \u02d9 Nk |G|2k + |X1||X2| . . . |Xk| Dk\u22121|G|k , where \u02d9 Nk = #{(x1, x2, . . . , xk, y1, y2, . . . , yk) \u2208(X1 \u00d7 X2 \u00d7 . . . \u00d7 Xk)2 : \u02d9 x1 \u02d9 x2 \u00b7 \u00b7 \u00b7 \u02d9 xk = \u02d9 y1 \u02d9 y2 \u00b7 \u00b7 \u00b7 \u02d9 yk}. For any X, Y \u2286G, the energies E(X, Y ) and \u02d9 E(X, Y ) are de\ufb01ned by E(X, Y ) := #{(x1, x2, y1, y2) \u2208X2 \u00d7 Y 2 : x1y1 = x2y2}, and \u02d9 E(X, Y ) := #{(x1, x2, y1, y2) \u2208X2 \u00d7 Y 2 : \u02d9 x1 \u02d9 y1 = \u02d9 x2 \u02d9 y2}. The next corollary follows directly from Theorem 1.4. Corollary 1.5. Let G = N \u22ca\u03d5 H be de\ufb01ned as previous, which is D-quasirandom on N. For any X, Y \u2286G, we have |H| |G| \u02d9 E(X, Y ) \u2264E(X, Y ) \u2264|H| |G| \u02d9 E(X, Y ) + |G| D |X||Y |. The next theorem provides an estimate of possible product growth. Theorem 1.6. Let G = N \u22ca\u03d5 H and \u02d9 Nk be de\ufb01ned as previous. Suppose that G is D-quasirandom on N. For an integer k \u22652 and subsets X1, X2, . . . , Xk of G, we have |X1X2 \u00b7 \u00b7 \u00b7 Xk| \u22651 2 min \u001a |N| \u02d9 Nk |X1|2|X2|2 \u00b7 \u00b7 \u00b7 |Xk|2, Dk\u22121 |G|k\u22121 |X1||X2| \u00b7 \u00b7 \u00b7 |Xk| \u001b . 4 \f1.2 Applications Let q be an odd prime power. We now take N0 = F2 q and H0 = SO2(Fq). De\ufb01ne the group homomorphism \u03d5 : H0 \u2192Aut(N0) by \u03d5(h) := \u03d5h with \u03d5h(x) = hx. Take the semi-direct product (G0, \u00b7) = (N0, +) \u22ca\u03d5 (H0, \u00b7). The set on the right-hand side is {(z, h) : z \u2208N0, h \u2208H0}, with the group law given by (z1, h1)(z2, h2) = (z1 + h1z2, h1h2). The group G0 consists of the rigid-motions that preserve \u2018distances\u2019 between pairs of points. For any (z, h) \u2208G0, this a\ufb03ne transformation is given by (z, h) : N0 \u2192N0 with (z, h)x = z + hx. The group H0 is given explicitly by H0 = (\" a \u2212b b a # : a2 + b2 = 1 ) . It is a cyclic group of order q \u2212\u03b5q, where \u03b5q = \u0012\u22121 q \u0013 = \uf8f1 \uf8f2 \uf8f3 1, if q \u22611 (mod 4), \u22121, if q \u22613 (mod 4). In the next result, we show that G0 is (q \u2212\u03b5q)-quasirandom on N0. Theorem 1.7. The group G0 has q \u2212\u03b5q type I irreducible complex representations of degree 1, and has q + \u03b5q type II irreducible complex representations of degree q \u2212\u03b5q. Remark 1.1. One can similarly consider the group G1 = F2 q \u22ca\u03d5 SL2(Fq), which is composed of a\ufb03ne maps that are area\u2013triangle\u2013invariant. It is not hard to prove that G1 is (q \u22121)/2quasirandom by using Theorem 1.7. Indeed, all complex representations of G1 of type II has degree no less than q \u2212\u03b5q, and all non-trivial representations of G1 of type I has dimension no less than (q \u22121)/2. Note that |G0| = (q \u2212\u03b5q)q2. Applying Theorems 1.3 and 1.6 on the group G0, and combining Theorem 1.7, we obtain the following two theorems immediately. Theorem 1.8. For any X, Y, Z \u2286G0, we have \f \f \f \f \f#{(x, y, z) \u2208X \u00d7 Y \u00d7 Z : xy = z} \u2212 \u02d9 M2 q2 \f \f \f \f \f \u2264q p |X||Y ||Z|, (3) 5 \fwhere \u02d9 M2 = #{(x, y, z) \u2208X \u00d7 Y \u00d7 Z : \u02d9 x \u02d9 y = \u02d9 z}. Theorem 1.9. For X, Y \u2286G0, we have |XY | \u226bmin \u001a q2|X|2|Y |2 #{(g1, g2, h1, h2) \u2208X2 \u00d7 Y 2 : \u02d9 g1 \u02d9 h1 = \u02d9 g2 \u02d9 h2} , |X||Y | q2 \u001b . (4) When the set X is close to a large subgroup of G, the error term q|X|3/2 in (3) and the original main term q\u22121|X|3 in (1) may be both smaller than the number of triples (x, y, z) \u2208X3 taken into consideration. So the re\ufb01nement of the main term in (3) is necessary. Moreover, the second term q\u22122|X|2 on the right-hand side of (4) may be larger than |XX|. So the replacement of the \ufb01rst term on the right-hand side of (2) by that of (4) is also necessary. The details are given in the following example, which also shows the sharpness of Theorem 1.9. Example 1.10. Let \u03b3 be a generator of the cyclic group H0. Assume that |H0| = q \u2212\u03b5q = kl, a = \u03b3k, and let X = {(t, aj) : t \u2208N0, 0 \u2264j \u2264l \u22121}. It is not hard to see that X is a subgroup of G. So, |XX| = |X| = lq2. and #{(x, y, z) \u2208X 3 : xy = z} = |X|2 = l2q4. The error term in (3) is q|X|3/2 = l3/2q4 \u2264l2q4, and the original main term in (1) is about q\u22123|X|3 = l3q3 \u2264l2q4. And the second term on the right-hand side of (4) is q\u22122|X|2 = l2q2 \u2265lq2. To calculate the quantities \u02d9 M2 = #{(x, y, z) \u2208X 3 : \u02d9 x \u02d9 y = \u02d9 z} and \u02d9 N2 = #{(g1, g2, h1, h2) \u2208X 4 : \u02d9 g1 \u02d9 h1 = \u02d9 g2 \u02d9 h2}, we see that for given x, y \u2208X, one has \u02d9 x \u02d9 y \u2208{aj : 0 \u2264j \u2264l \u22121} and #{z \u2208X : \u02d9 z = \u02d9 x \u02d9 y} = |N0|. It follows that \u02d9 M2 = |X|2|N0|. Similarly, \u02d9 N2 = |X|3|N0|. Thus, Theorems 1.8 and 1.9 show that |l2q4 \u2212l2q4| \u2264l3/2q4 and lq2 \u226bmin{lq2, l2q2}. In particular, Theorem 1.9 is optimal in this example. Remark 1.2. If the reader is interested in analogs of Theorems 1.8 and 1.9 in the setting of the group G1 = F2 q \u22ca\u03d5 SL2(Fq), then similar results can be derived in the same way by using Remark 1.1. Denote by \u03c10, \u03c11, . . . , \u03c1q\u2212\u03b5q\u22121 the irreducible representations of G0 of type I. When X = Y , the next theorem o\ufb00ers a lower bound in terms of the Fourier bias of the set X. 6 \fTheorem 1.11. For X \u2286G0, assume that |c 1X(\u03c1r)| \u2264M for all but k indices r \u2208{0, 1, 2, . . . , q \u2212\u03b5q \u22121}. Then |XX| \u226bmin \u001aq3 k , |X|4 q10M4 , |X|2 q2 \u001b . When M = 0, the second term on the right-hand side of above formula can be omitted. Example 1.10 also shows that this theorem is sharp. The details are given in Example 5.1. In the next theorem, we prove that if the set X satis\ufb01es certain properties, then the product XX grows exponentially. For two points x = (x1, x2) and y = (y1, y2) in F2 q, de\ufb01ne \u2225x \u2212y\u2225= (x1 \u2212y1)2 + (x2 \u2212y2)2. We say that (x, y) \u2208F2 q is a line segment of length t if \u2225x \u2212y\u2225= t. Note that rigid motions map line segments to line segments of same length. Now let E be a given subset of F2 q with |E| > q3/2. For t \u0338= 0, let nt be the number of line segments (x, y) \u2208E \u00d7 E such that ||x \u2212y|| = t. Iosevich and Rudnev [7] proved that nt = (1 + o(1))|E|2/q, here o(1) \u21920 as q \u2192\u221e. Fix a line segment (u0, v0) \u2208E \u00d7 E of length t. For each (x, y) \u2208E \u00d7 E such that ||x \u2212y|| = t, there exists a unique rigid motion (z, \u03b8) \u2208G0 such that (z, \u03b8)x = u0 and (z, \u03b8)y = v0. Let Xt be the set of all corresponding rigid motions when (x, y) \u2208E \u00d7 E runs over all pairs of length t. Then |Xt| = nt = (1 + o(1))|E|2/q. Theorem 1.12. Let E \u2286F2 q with |E| = q\u03b1 and \u03b1 \u2208(3/2, 2). Let t \u2208F\u2217 q. Then there exists \u03b4 = \u03b4(\u03b1) > 0 such that |XtXt| \u226b|Xt|1+\u03b4. In particular, we have |XtXt| \u226bmin \u001a q|E|, |E|4 q4 \u001b . In the above theorem, |XtXt| \u226bq3 if and only if |E| \u226bq2. This raises the following question: does there exist \u01eb > 0 such that |XtXt| \u226bq3 whenever |E| \u226bq2\u2212\u01eb? 2 Preliminaries In this section, we recall some basic properties of semi-driect products and non-abelian Fourier analysis. 7 \fIn G = N \u22ca\u03d5 H, the identity element is (1N, 1H), and the inverse is given by (z, h)\u22121 = (\u03d5h\u22121(z\u22121), h\u22121). The product of two elements (z1, h1) and (z2, h2) is determined by (z1, h1)(z2, h2) = (z1 \u03d5h1(z2), h1h2). Recall the notation g := (\u00a8 g, \u02d9 g) = (g\u00b7\u00b7, g\u00b7) for g \u2208G. It follows from g1g2 = ( \u00a8 g1, \u02d9 g1)( \u00a8 g2, \u02d9 g2) = ( \u00a8 g1 \u03d5 \u02d9 g1( \u00a8 g2), \u02d9 g1 \u02d9 g2), g\u22121 = (\u00a8 g, \u02d9 g)\u22121 = (\u03d5\u02d9 g\u22121(\u00a8 g\u22121), \u02d9 g\u22121) that (g1g2)\u00b7 = \u02d9 g1 \u02d9 g2 and (g\u22121)\u00b7 = (\u02d9 g)\u22121. Let b G and b H be the sets of all (non-isomorphic) irreducible complex representations of G and H, respectively. Let Q and Q\u2032 be positive integers. Assume that b H = { \u02d9 \u03c10, \u02d9 \u03c11, . . . , \u02d9 \u03c1Q\u22121}, here \u02d9 \u03c10 is the trivial representation. Then all irreducible representations of G of type I is given by b GI = {\u03c10, \u03c11, . . . , \u03c1Q\u22121}, where \u03c1r(g) = \u02d9 \u03c1r(\u02d9 g), (g \u2208G, 0 \u2264r \u2264Q \u22121). Here \u03c10 is the trivial representation. We also denote the set of the irreducible representations of G of type II by b GII = {\u03c1\u2032 1, \u03c1\u2032 2, . . . , \u03c1\u2032 Q\u2032}. Therefore, b G = {\u03c10, . . . , \u03c1Q\u22121, \u03c1\u2032 1, . . . , \u03c1\u2032 Q\u2032}. For each representation \u03c1, its degree will be denoted by d\u03c1, and its character, denoted by \u03c7\u03c1 : G \u2192 C, is de\ufb01ned by \u03c7\u03c1(x) = Tr(\u03c1(x)). The main tool is non-abelian Fourier analysis. For each representation (\u03c1, V ) with \u03c1 \u2208b G, we may assume that the inner product on V is chosen so that \u03c1 is unitary. Assume that the inner product and norm on V is given by < a, b >HS= Tr(b\u2217a), \u2225a\u22252 HS = \u27e8a, a\u27e9HS = Tr(a\u2217a). It satis\ufb01es the properties that |\u27e8a, b\u27e9|HS \u2264\u2225a\u2225HS\u2225b\u2225HS, and \u2225ab\u2225HS \u2264\u2225a\u2225HS\u2225b\u2225HS. For a function f : G \u2192C, we use the notation. Ex\u2208Gf(x) = 1 |G| X x\u2208G f(x). 8 \fThe Fourier transformation is de\ufb01ned by b f(\u03c1) = Ex\u2208Gf(x)\u03c1(x), (\u03c1 \u2208b G), and the Fourier inverse is given by f(x) = X \u03c1\u2208b G d\u03c1\u27e8b f(\u03c1), \u03c1(x)\u27e9HS, (x \u2208G). The Parseval\u2019s identity is \u27e8f1, f2\u27e9= X \u03c1\u2208b G d\u03c1\u27e8b f1(\u03c1), b f2(\u03c1)\u27e9HS, where the inner product of two functions f and g on G, denoted by \u27e8f, g\u27e9, is \u27e8f, g\u27e9= Ex\u2208Gf(x)g(x). The convolution of two functions f1 and f2 on G is de\ufb01ned by (f1 \u2217f2)(x) = Ey\u2208Gf1(xy\u22121)f2(y). We have the property that \\ f1 \u2217f2 = b f1 b f2. For p \u22651, the lp-norm of f is given by \u2225f\u2225p = (Ex\u2208G|f(x)|p)1/p . If 1/r = 1/p + 1/q, then we have the H\u00a8 older inequality \u2225fg\u2225r \u2264\u2225f\u2225p\u2225g\u2225q. For a subset X \u2286G, one has \u22251X\u22251 = \u22251X\u22252 2 = |X|/|G|. 3 Proof of Theorems 1.3, 1.4, and 1.6 Proof of Theorem 1.3. Note that the characteristic functions 1Xj (0 \u2264j \u2264k) all take non-negative value. The number of solutions is counted by 1 |G|k #{(x0, x1, . . . , xk) \u2208X0 \u00d7 X1 \u00d7 . . . \u00d7 Xk : x1x2 . . . xk = x0} = Ex\u2208G(1X1 \u22171X2 \u2217. . . \u22171Xk)(x)1X0(x) = \u27e81X1 \u22171X2 \u2217. . . \u22171Xk, 1X0\u27e9 = X \u03c1\u2208b G d\u03c1  (1X1 \u22171X2 \u2217. . . \u22171Xk) V (\u03c1), d 1X0(\u03c1) \u000b HS, where the Parseval\u2019s identity is applied. Now we split the sum into two parts: M = Q\u22121 X r=0 d\u03c1r\u27e8(1X1 \u22171X2 \u2217. . . \u22171Xk) V (\u03c1r), d 1X0(\u03c1r)\u27e9HS, E = Q\u2032 X j=1 d\u03c1\u2032 j\u27e8(1X1 \u22171X2 \u2217. . . \u22171Xk) V (\u03c1\u2032 j), d 1X0(\u03c1\u2032 j)\u27e9HS. 9 \fFor type II representions, we have |E| \u2264 Q\u2032 X j=1 d\u03c1\u2032 j\u2225(1X1 \u22171X2 \u2217. . . \u22171Xk) V (\u03c1\u2032 j)\u2225HS \u2225d 1X0(\u03c1\u2032 j)\u2225HS \u2264 \uf8eb \uf8ed Q\u2032 X j=1 d\u03c1\u2032 j\u2225(1X1 \u22171X2 \u2217. . . \u22171Xk) V (\u03c1\u2032 j)\u22252 HS \uf8f6 \uf8f8 1/2 \uf8eb \uf8ed Q\u2032 X j=1 d\u03c1\u2032 j\u2225d 1X0(\u03c1\u2032 j)\u22252 HS \uf8f6 \uf8f8 1/2 . Since d\u03c1\u2032 j \u2265D (1 \u2264j \u2264Q\u2032), one has Q\u2032 X j=1 d\u03c1\u2032 j\u2225(1X1 \u22171X2 \u2217. . . \u22171Xk) V (\u03c1\u2032 j)\u22252 HS = Q\u2032 X j=1 d\u03c1\u2032 j\u2225d 1X1(\u03c1\u2032 j)d 1X2(\u03c1\u2032 j) . . . d 1Xk(\u03c1\u2032 j)\u22252 HS \u2264 1 Dk\u22121 Q\u2032 X j=1 dk \u03c1\u2032 j\u2225d 1X1(\u03c1\u2032 j)\u22252 HS \u2225d 1X2(\u03c1\u2032 j)\u22252 HS . . . \u2225d 1Xk(\u03c1\u2032 j)\u22252 HS \u2264 1 Dk\u22121 \uf8eb \uf8ed Q\u2032 X j=1 d\u03c1\u2032 j\u2225d 1X1(\u03c1\u2032 j)\u22252 HS \uf8f6 \uf8f8 \uf8eb \uf8ed Q\u2032 X j=1 d\u03c1\u2032 j\u2225d 1X2(\u03c1\u2032 j)\u22252 HS \uf8f6 \uf8f8. . . \uf8eb \uf8ed Q\u2032 X j=1 d\u03c1\u2032 j\u2225d 1Xk(\u03c1\u2032 j)\u22252 HS \uf8f6 \uf8f8 \u2264 1 Dk\u22121 \uf8eb \uf8edX \u03c1\u2208b G d\u03c1\u2225d 1X1(\u03c1)\u22252 HS \uf8f6 \uf8f8 \uf8eb \uf8edX \u03c1\u2208b G d\u03c1\u2225d 1X2(\u03c1)\u22252 HS \uf8f6 \uf8f8. . . \uf8eb \uf8edX \u03c1\u2208b G d\u03c1\u2225d 1Xk(\u03c1)\u22252 HS \uf8f6 \uf8f8 = 1 Dk\u22121 \u22251X1\u22252 2\u22251X2\u22252 2 . . . \u22251Xk\u22252 2 = |X1||X2| . . . |Xk| Dk\u22121|G|k . Similarly, Q\u2032 X j=1 d\u03c1\u2032 j\u2225d 1X0(\u03c1\u2032 j)\u22252 HS \u2264\u22251X0\u22252 2 = |X0| |G| . Thus, E \u2264 s |X0||X1| . . . |Xk| Dk\u22121|G|k+1 . For type I represetations, we have M = Q\u22121 X r=0 d\u03c1r\u27e8(1X1 \u22171X2 \u2217. . . \u22171Xk) V (\u03c1r), d 1X0(\u03c1r)\u27e9HS = Q\u22121 X r=0 d\u03c1rEu\u2208GEv\u2208G(1X1 \u22171X2 \u2217. . . \u22171Xk)(u)1X0(v)\u27e8\u03c1r(u), \u03c1r(v)\u27e9HS = Eu\u2208GEv\u2208G(1X1 \u22171X2 \u2217. . . \u22171Xk)(u)1X0(v) Q\u22121 X r=0 d\u03c1r\u03c7\u03c1r(v\u22121u). 10 \fFor given u, v \u2208G, we have (v\u22121u)\u00b7 = \u02d9 v\u22121 \u02d9 u. Then Q\u22121 X r=0 d\u03c1r\u03c7\u03c1r(v\u22121u) = Q\u22121 X r=0 d \u02d9 \u03c1r\u03c7 \u02d9 \u03c1r( \u02d9 v\u22121 \u02d9 u) = X \u03c1\u2208b H d\u03c1\u03c7\u03c1( \u02d9 v\u22121 \u02d9 u) = \uf8f1 \uf8f2 \uf8f3 |H|, if \u02d9 u = \u02d9 v, 0, if \u02d9 u \u0338= \u02d9 v. Now we have M = |H| |G|2 X u,v\u2208G \u02d9 u= \u02d9 v (1X1 \u22171X2 \u2217. . . \u22171Xk)(u)1X0(v) = |H| |G|k+1 #{(x0, x1, . . . , xk) \u2208X0 \u00d7 X1 \u00d7 . . . \u00d7 Xk : \u02d9 x1 \u02d9 x2 . . . \u02d9 xk = \u02d9 x0}. Combining all above formulae, the Theorem 1.3 then follows. Proof of Theorem 1.4. Note that \u22251X1 \u22171X2 \u2217. . . \u22171Xk\u22252 2 = X \u03c1\u2208b G d\u03c1\u2225(1X1 \u22171X2 \u2217. . . \u22171Xk) V (\u03c1r)\u22252 HS = X \u03c1\u2208b G d\u03c1\u2225d 1X1(\u03c1)d 1X2(\u03c1) . . . d 1Xk(\u03c1)\u22252 HS. We use similar argument as in the proof of Theorem 1.3. Let M = Q\u22121 X r=0 d\u03c1r\u2225(1X1 \u22171X2 \u2217. . . \u22171Xk) V (\u03c1r)\u22252 HS, E = Q\u2032 X j=1 d\u03c1\u2032 j\u2225d 1X1(\u03c1\u2032 j)d 1X2(\u03c1\u2032 j) . . . d 1Xk(\u03c1\u2032 j)\u22252 HS. For type II representions, we have 0 \u2264E \u2264 1 Dk\u22121 Q\u2032 X j=1 \u0010 d\u03c1\u2032 j\u2225d 1X1(\u03c1\u2032 j)\u22252 HS \u0011 \u0010 d\u03c1\u2032 j\u2225d 1X2(\u03c1\u2032 j)\u22252 HS \u0011 . . . \u0010 d\u03c1\u2032 j\u2225d 1Xk(\u03c1\u2032 j)\u22252 HS \u0011 \u2264 1 Dk\u22121 \uf8eb \uf8ed Q\u2032 X j=1 d\u03c1\u2032 j\u2225d 1X1(\u03c1\u2032 j)\u22252 HS \uf8f6 \uf8f8 \uf8eb \uf8ed Q\u2032 X j=1 d\u03c1\u2032 j\u2225d 1X2(\u03c1\u2032 j)\u22252 HS \uf8f6 \uf8f8. . . \uf8eb \uf8ed Q\u2032 X j=1 d\u03c1\u2032 j\u2225d 1Xk(\u03c1\u2032 j)\u22252 HS \uf8f6 \uf8f8 \u2264 1 Dk\u22121 \u22251X1\u22252 2\u22251X2\u22252 2 . . . \u22251Xk\u22252 2 = |X1||X2| . . . |Xk| Dk\u22121|G|k . 11 \fFor type I represetations, we have M = Q\u22121 X r=0 d\u03c1r\u27e8(1X1 \u22171X2 \u2217. . . \u22171Xk) V (\u03c1r), (1X1 \u22171X2 \u2217. . . \u22171Xk) V (\u03c1r)\u27e9HS = Q\u22121 X r=0 d\u03c1rEu\u2208GEv\u2208G(1X1 \u22171X2 \u2217. . . \u22171Xk)(u)(1X1 \u22171X2 \u2217. . . \u22171Xk)(v)\u27e8\u03c1r(u), \u03c1r(v)\u27e9HS = Eu\u2208GEv\u2208G(1X1 \u22171X2 \u2217. . . \u22171Xk)(u)(1X1 \u22171X2 \u2217. . . \u22171Xk)(v) Q\u22121 X r=0 d\u03c1r\u03c7r(v\u22121u) = |H| |G|2 X u,v\u2208G \u02d9 u= \u02d9 v (1X1 \u22171X2 \u2217. . . \u22171Xk)(u)(1X1 \u22171X2 \u2217. . . \u22171Xk)(v) = |H| |G|2k #{(x1, x2, . . . , xk, y1, y2, . . . , yk) \u2208(X1 \u00d7 X2 \u00d7 . . . \u00d7 Xk)2 : \u02d9 x1 \u02d9 x2 . . . \u02d9 xk = \u02d9 y1 \u02d9 y2 . . . \u02d9 yk}. The theorem now follows. The proof of Theorem 1.6 is based on the following lemma. Lemma 3.1. Let G be a group and Z \u2286G. Suppoes that f : G \u2192C is a function whose support is contained in Z. Then |Z| \u2265|G|\u2225f\u22252 1 \u2225f\u22252 2 Proof. By Cauchy-Schwartz inequality, we have \u2225f\u22252 1 = (Ex\u2208G|f(x)|)2 =   1 |G| X x\u2208Z |f(x)| !2 \u2264   1 |G| X x\u2208Z 12 !   1 |G| X x\u2208Z |f(x)|2 ! = |Z| |G|\u2225f\u22252 2. The proof is completed. Proof of Theorem 1.6. Note that the support of 1X1 \u22171X2 \u2217. . . \u22171Xk is exactly X1X2 . . . Xk. By Lemma 3.1, we have |X1X2 . . . Xk| \u2265|G|\u22251X1 \u22171X2 \u2217. . . \u22171Xk\u22252 1 \u22251X1 \u22171X2 \u2217. . . \u22171Xk\u22252 2 . The l1-norm can be computed directly by \u22251X1 \u22171X2 \u2217. . . \u22171Xk\u22251 = 1 |G|k X x\u2208G X x1,...,xk\u2208G x1x2...xk=x 1X1(x1)1X2(x2) . . . 1Xk(xk) = |X1||X2| . . . |Xk| |G|k . (5) Combing Theorem 1.4, Theorem 1.6 then follows. 4 Proof of Theorem 1.7 In the rest of this paper, we always write Q = q \u2212\u03b5q = |H0| and Q\u2032 = q + \u03b5q. Then QQ\u2032 = q2 \u22121. 12 \fFor G0 = N0 \u22ca\u03d5 H0 with N0 = F2 q and H0 = SO2(Fq), the group operation is given by (z1, h1)(z2, h2) = (z1 + h1z2, h1h2). We also have the identity element (0, 1), where 0 is the zero vector in N0 and 1 is the identity matrix in H, and the inverse of (z, h) is (z, h)\u22121 = (\u2212h\u22121z, h\u22121). Recall that H0 is abelien, a conjugation of elements is given by (w, k)(z, h)(w, k)\u22121 = (w + kz, kh)(\u2212k\u22121w, k\u22121) = (kz + (1 \u2212h)w, h). Lemma 4.1. For any h \u2208H0 with h \u0338= 1, the matrix 1 \u2212h is ivertible. Proof. Let h = \" a \u2212b b a # , where a2 + b2 = 1. We have det(1 \u2212h) = \f \f \f \f \f 1 \u2212a b \u2212b 1 \u2212a \f \f \f \f \f = 1 \u22122a + a2 + b2 = 2(1 \u2212a). Since h is not the identity matrix, we have a \u0338= 1 and det(1 \u2212h) \u0338= 0. So 1 \u2212h is invertible. Recall that \u03d5 : H0 \u2192Aut(N0) is given by \u03d5h(x) = hx. By Lemma 4.1, we have StabH0(x) = {h \u2208H0 : hx = x} = {1} for x \u2208N0 \\ {0}. It follows that the cardinality of the orbit of x is |OrbH0(x)| = |H0| |StabH0(x)| = Q. (6) Indeed, apart from the orbit of 0 with length one, there are Q\u2032 disjoint orbits with length Q. It saties\ufb01es that |N0| = q2 = 1 + QQ\u2032. Proof of Theorem 1.7. The cardinality of G0 is |G0| = Qq2. Since H0 is an cyclic group, the type I irreducible representations of G0 are lifted from representations of H0, and all have degree 1. More explicitly, let \u03b3 be a generator of H0, i.e., H0 = {\u03b3j : 0 \u2264j \u2264Q \u22121}, then the representations can be given by \u03c1r \u0000(z, \u03b3j) \u0001 = \u02d9 \u03c1r(\u03b3j) = e2\u03c0i rj Q , (z \u2208N, 0 \u2264j \u2264Q \u22121). (7) Moreover, since f N0 = {(z, 1) : z \u2208N} is an abelian subgroup of G0 with [G0 : f N0] = |H0|, the dimension of any irreducible representation of G0 is no larger than |H0|. In the following, we will \ufb01nd all conjugacy classes of G0 and then determine the degree of type II 13 \frepresentations. For (z, 1) \u2208G0 with z \u0338= 0, (w, k)(z, 1)(w, k)\u22121 = (kz, 1) for any (w, k) \u2208G0. Here kz \u0338= 0, since k is invertible. Now the conjugacy class of (z, 1) is [(z, 1)] = {(kz, 1) : k \u2208H0}. It follows from (6) that #[(z, 1)] = |OrbH0(z)| = Q. Indeed, the set f N0 \\ {(0, 1)} is divided into Q\u2032 disjoint conjugacy classes, each of cardinality Q. Moreover, [(0, 1)] = {(0, 1)} is another conjugacy class. Next we consider the conjugacy class of (0, h) for some h \u2208H0 \\ {1}. One has (w, k)(0, h)(w, k)\u22121 = \u0000(1 \u2212h)w, h \u0001 for any (w, k) \u2208G0. Since 1 \u2212h is invertible by Lemma 4.1, one has [(0, h)] = {(y, h) : y \u2208N0}. There are Q \u22121 such conjugacy classes. Now, let \u03c10, \u03c1r (r = 1, 2, . . . , q \u2212\u03b5q \u22121) and \u03c1\u2032 j (j = 1, 2, . . . , Q\u2032) be the irreducible representations of G0 corresponding to the conjugacy classes [(0, 1)], [(0, h)] (h \u0338= 1) and [(z, 1)] (z \u0338= 0). Then Q2Q\u2032 \u2265 Q\u2032 X j=1 d2 \u03c1\u2032 i = |G|2 \u2212 Q\u22121 X r=0 d2 \u03c1r = Qq2 \u2212Q = Q2Q\u2032. It follows that the equality holds, i.e., d\u03c1\u2032 i = Q for all j = 1, 2, . . . , Q\u2032. The proof is completed. 5 Proof of Theorems 1.11 and 1.12 Proof of Theorem 1.11. Recall that Q = q\u2212\u01ebq. For any 0 \u2264r \u2264Q\u22121, we have |c 1X(\u03c1r)| \u2264\u22251X\u22251 = |X|/|G0|. Assume that {0, 1, . . . , Q \u22121} = \u03930 \u2294\u03931, where |\u03930| = k and supr\u2208\u03931 |c 1X(\u03c1r)| \u2264M. We now split the sums \u22251X \u22171X\u22252 2 = X \u03c1\u2208c G0 d\u03c1\u2225\\ 1X \u22171X(\u03c1)\u22252 HS = X \u03c1\u2208c G0 d\u03c1\u2225c 1X(\u03c1)2\u22252 HS into three parts \u03a30 = X r\u2208\u03930 d\u03c1r|c 1X(\u03c1r)|4 \u2264k |X|4 |G0|4 , \u03a31 \u2264 X r\u2208\u03931 d\u03c1r|c 1X(\u03c1r)|4 \u2264(Q \u2212k)M4, 14 \fand \u03a32 = Q\u2032 X j=1 d\u03c1\u2032 j\u2225c 1X(\u03c1\u2032 j)2\u22252 HS \u22641 D Q\u2032 X j=1 d2 \u03c1\u2032 j\u2225c 1X(\u03c1\u2032 j)\u22252 HS\u2225c 1X(\u03c1\u2032 j)\u22252 HS \u22641 D \uf8eb \uf8ed Q\u2032 X j=1 d\u03c1\u2032 j\u2225c 1X(\u03c1\u2032 j)\u22252 HS \uf8f6 \uf8f8 2 \u22641 D \u0000\u22251X\u22252 2 \u00012 = 1 D |X|2 |G0|2 . Thus, \u22251X \u22171X\u22252 2 \u2264k |X|4 |G0|4 + (Q \u2212k)M4 + 1 D |X|2 |G0|2 , Using Theorem 1.7, Lemma 3.1 and (5), one obtains |XX| \u226bmin \u001a|G0| k , |X|4 |G0|3(Q \u2212k)M4 , D|X|2 |G0| \u001b \u226bmin \u001aq3 k , |X|4 q10M4 , |X|2 q2 \u001b . The theorem then follows. Example 5.1. Let X be the set chosen in Example 1.10. Recall that Q = q \u2212\u03b5q = kl. For 0 \u2264r \u2264Q \u22121, by (7) we have c 1X (\u03c1r) = Ex\u2208G1X (x)\u03c1r(x) = 1 |G0| X z\u2208N l\u22121 X j=0 e2\u03c0i rk Q j = 1 |G0| X z\u2208N l\u22121 X j=0 e2\u03c0i r l j = \uf8f1 \uf8f2 \uf8f3 1 k, if l|r, 0, if l \u2224r. As a result, in Theorem 1.11, we can take M = 0 for all but Q/l = k irreducible representations of dimension 1, and obtain q3 k = |X| = |XX| \u226bmin \u001aq3 k , q4 k2 \u001b \u226bq3 k , which gives the correct order. Proof of Theorem 1.12. We \ufb01rst start with the estimate that #{(g1, g2, h1, h2) \u2208X4 t : \u02d9 g1 \u02d9 h1 = \u02d9 g2 \u02d9 h2} \u2264|Xt|3|E|. Indeed, if we \ufb01x g1, g2, and h1, then the number of h2 is at most the size of E. Thus, it follows from Theorem 1.9 that |XtXt| \u226bmin \u001a q2|Xt|4 |Xt|3|E|, |Xt|2 q2 \u001b \u226bmin{q|E|, q\u22124|E|4} \u226bqmin{1+\u03b1,4(\u03b1\u22121)} \u226bqmin{2\u2212\u03b1,2\u03b1\u22123}|Xt|. That is to say, we can take \u03b4(\u03b1) = min{2 \u2212\u03b1, 2\u03b1 \u22123} > 0. The proof is completed. 15 \f6 Acknowledgements The authors would like to thank Junbin Dong for helpful discussions on the representation theory. The \ufb01rst author would like to thank the Vietnam Institute for Advanced Study in Mathematics (VIASM) for the hospitality and for the excellent working condition. The second author is supported by funds provided by ShanghaiTech University."
+                },
+                {
+                    "url": "http://arxiv.org/abs/2304.10897v3",
+                    "title": "Triangles with one fixed side-length, a Furstenberg type problem, and incidences in finite vector spaces",
+                    "abstract": "The first goal of this paper is to prove a sharp condition to guarantee of\nhaving a positive proportion of all congruence classes of triangles in given\nsets in $\\mathbb{F}_q^2$. More precisely, for $A, B, C\\subset \\mathbb{F}_q^2$,\nif $|A||B||C|^{1/2}\\gg q^4$, then for any $\\lambda\\in \\mathbb{F}_q\\setminus\n\\{0\\}$, the number of congruence classes of triangles with vertices in $A\\times\nB\\times C$ and one side-length $\\lambda$ is at least $\\gg q^2$. In higher\ndimensions, we obtain similar results for $k$-simplex but under a slightly\nstronger condition. Compared to the well--known $L^2$ method in the literature,\nour approach offers better results in both conditions and conclusions. When\n$A=B=C$, the second goal of this paper is to give a new and unified proof of\nthe best current results on the distribution of simplex due to Bennett, Hart,\nIosevich, Pakianathan, and Rudnev (2017) and McDonald (2020). The third goal of\nthis paper is to study a Furstenberg-type problem associated to a set of rigid\nmotions. The main ingredients in our proofs are incidence bounds between points\nand rigid motions. While the incidence bounds for large sets are due to the\nauthor and Semin Yoo (2023), the bound for small sets will be proved by using a\npoint--line incidence bound in $\\mathbb{F}_q^3$ due to Koll\\'{a}r (2015).",
+                    "authors": "Thang Pham",
+                    "published": "2023-04-21",
+                    "updated": "2023-08-15",
+                    "primary_cat": "math.CO",
+                    "cats": [
+                        "math.CO",
+                        "math.NT"
+                    ],
+                    "main_content": "Introduction 2 2 Incidences between points and rigid motions 5 3 Triangles with one \ufb01xed side-length (Theorem 1.2) 7 4 Extension in higher dimensions for simplex (Theorem 1.4) 9 5 Rich rigid motions and simplices (Theorems 1.5 and 1.6) 9 6 Furstenberg type problem for rigid motions (Theorems 1.9, 1.10, 1.11, and 1.12) 13 7 Incidences for small sets (Theorem 2.8) 13 8 Discussions over prime \ufb01elds 16 9 Acknowledgements 18 \u2217University of Science, Vietnam National University, Hanoi. Email: phamanhthang.vnu@gmail.com 1 \f1 Introduction Let Fq be a \ufb01nite \ufb01eld of order q, where q is a prime power. Let x = (x1, . . . , xk+1) and y = (y1, . . . , yk+1) be two k-simplices in Fd q. We say that these two simplices are in the same congruence class, denoted by (x1, . . . , xk+1) \u223c(y1, . . . , yk+1), if there exist g \u2208O(d) and z \u2208Fd q such that gxi + z = yi for all 1 \u2264i \u2264k + 1. For each ksimplex x = (x1, . . . , xk+1), by dim(x) we mean the dimension of the space spanned by vectors {x2 \u2212x1, . . . , xk+1 \u2212x1}. The k-simplex x is called non-degenerate if dim(x) = min{k, d} and degenerate otherwise. Let A1, . . . , Ak+1 be subsets of Fd q, we say the k-simplex x = (x1, . . . , xk+1) has vertices in Qk+1 i=1 Ai if xi \u2208Ai for all 1 \u2264i \u2264k + 1. In this paper, we study the following question. Question 1.1. Let 2 \u2264k \u2264d be an integer, and A1, . . . , Ak+1 be subsets of Fd q. Given a (k \u22121)non-degenerate simplex \u2206k\u22121, what conditions on Ai do we need to ensure that the number of congruence classes of k-simplices with vertices in Qk+1 i=1 Ai containing a copy of \u2206k\u22121 is at least \u226bqk? Our \ufb01rst result is on the case d = k = 2. Theorem 1.2. Let A, B, C be sets in F2 q with q \u22613 mod 4. Assume that |A||B||C|1/2 \u226bq4 then, for any \u03bb \u2208Fq \\{0}, the number of congruence classes of triangles with vertices in A\u00d7B \u00d7C and one edge of length \u03bb is at least \u226bq2. As a consequence, if |A||B||C|1/2 \u226bq4, then the number of congruence classes of triangles with vertices in A \u00d7 B \u00d7 C is at least \u226bq3. This theorem is sharp in the sense that the lower bound q4 can not be improved to q4\u2212\u01eb for any \u01eb > 0. In particular, for any \u01eb > 0, there exist A, B, C \u2282F2 q with |A||B||C|1/2 \u223cq4\u22122\u01eb such that the number of congruence classes of triangles with vertices in A\u00d7B \u00d7C is at most q3\u2212\u01eb. A detailed construction will be provided in Section 3. The study of this type of question was introduced by Furstenberg, Katznelson and Weiss [6], namely, they proved that for A \u2282R2, if A has positive upper Lebesgue density, then the \u03b4neighborhood of A, for any \u03b4 > 0, contains a congruence copy of a large dilate of every three points con\ufb01guration. We note that taking the \u03b4-neighborhood of A is necessary, this comes from an example of Bourgain [2] with the three points forming an arithmetic progression. Bourgain also extended this result for any k-simplex with k < d. In the integer lattice Zd, Magyar [12, 13] studied similar questions, in particular, he showed that if A \u2282Zd, d > 2k + 4, has positive upper density, then A contains all large dilates of a given k-simplex. We refer the interested reader to [12, 13, 11] and references therein for more discussions and recent results in this direction. In the setting of \ufb01nite \ufb01elds, the \ufb01rst result on Question 1.1 was given by Bennett, Iosevich, and Pakianathan [3] in 2014. They proved the following theorem. Theorem 1.3 ([3]). Let A be a set in F2 q with q \u22613 mod 4. Assume that |A| \u226bq7/4, then for any \u03bb \u2208Fq \\{0}, the number of congruence classes of triangles with vertices in A \u00d7 A \u00d7 A and one edge of length \u03bb is at least \u226bq2. To compare with Theorem 1.2, assume that A = B = C, then we will need the condition that |A| \u226bq8/5. This improves the exponent 7/4 from the previous result. We now sketch the main idea for this improvement. For a rigid motion r = (g, z) \u2208O(2) \u00d7 F2 q, we say that the point 2 \f(u, v) \u2208F2 q \u00d7 F2 q is incident to r if \u03c6r(u) := gu + z = v. When |A| is large enough, say |A| \u226bq3/2, then the number of pairs (u, v) \u2208A \u00d7 A such that ||u \u2212v|| = t, t \u0338= 0, is about |A|2/q. Fix one of those pairs, say (u0, v0), for each pair (u, v) with ||u \u2212v|| = t, there exists unique a rigid motion r = (g, z) that maps u to u0 and v to v0. We denote the set of corresponding rigid motions by R. The proof is now reduced to a Furstenberg type problem of showing that the set [ r\u2208R \u03c6r(A) (1) has size of at least \u226bq2. To bounding the size of this set, we will need a point-rigid motion incidence bound. Compared to the proof in [3], there are two new perspectives in our proof: a stronger incidence theorem (Theorem 2.4) due to the author and Yoo [16], and a more e\ufb00ective mechanism to apply this theorem. We want to make a remark here that in both Theorems 1.2 and 1.3, the condition q \u22613 mod 4 is required. This is needed in the proofs of incidence bounds. It would be interesting to see if the same result holds when q \u22611 mod 4. The same approach can be applied for the problem of k-simplex in higher dimensions, but with a stronger condition. Theorem 1.4. Given a (k \u22121)-non-degenerate simplex \u2206k\u22121 with nonzero side-lengths in Fd q. Assume Qk+1 i=1 |Ai| \u226bqdk+1 and |Ai| \u226bq d\u22121 2 +k\u22121 for all 1 \u2264i \u2264k, then the number of congruence classes of k-simplices containing a copy of \u2206k\u22121 with vertices in Qk+1 i=1 Ai is at least \u226bqk. When d = 2 and k = 2, the above theorem tells us that the three sets A, B, C have to satisfy |A||B||C| \u226bq5, which is worse than the condition |A||B||C|1/2 \u226bq4 of Theorem 1.2. This is because the proof of Theorem 1.4 involves a more general incidence theorem which is not very strong in some speci\ufb01c dimensions. One more remark we want to add here is that the condition q \u22613 mod 4 is not required in this theorem. If we only count the number of congruence classes of k-simplex in a given set, the next theorem due to Bennett, Hart, Iosevich, Pakianathan, and Rudnev [4], and McDonald [14]. Theorem 1.5 ([4, 14]). Let A be a set in Fd q. 1. If 1 \u2264k \u2264d and |A| \u226bq dk+1 k+1 , then the number of congruence classes of non-degenerate k-simplex in A is at least \u226bq(k+1 2 ). 2. If k > d and |A| \u226bq dk+1 k+1 , then the number of congruence classes of non-degenerate k-simplex in A is at least qd(k+1)\u2212(d+1 2 ). The main idea in the proof of this theorem is to bound the L2-norm of k-simplices. In this paper, we present a uni\ufb01ed proof using results on the number of k-rich rigid motions. This approach also leads to the following improvement in two dimensions. Theorem 1.6. For A \u2282F2 q with |A| \u226bq 4k 2k+1 and q \u22613 mod 4, we have the number of congruence classes of non-degenerate k-simplex in A is at least q2k\u22121. Unlike Theorem 1.2, we do not have any constructions on the sharpness of Theorems 1.5 and 1.6. For k \u2264d, let \u03b1k,d be the smallest exponent such that the number of congruence classes of k-simplex in A is at least \u226bq(k+1 2 ) whenever |A| \u226bq\u03b1k,d. The best current lower bounds of \u03b1k,d are recorded in the following theorem. 3 \fTheorem 1.7 ([4]). Let k and d be integers with k \u2264d. i. If k = 1 and d \u22653 odd, we have \u03b11,d \u2265d+1 2 . ii. If k = 1 and d \u22652 is even, we have \u03b11,d \u2265d 2. iii. If k > 1, we have \u03b1k,d \u2265k \u22121 + 1 k. Assume k \u2264d, if Qk+1 i=1 |Ai| \u226bqdk+1, then, by Theorem 1.4, the number of congruence classes of k-simplices in with vertices in Qk+1 i=1 Ai is at least \u226bq(k+1 2 ). One might ask if the method in the proof of Theorem 1.5 also works for di\ufb00erent sets. The answer is positive, but stronger assumptions are needed. For instance, when k = 2, we will need |A1|, |A2|, |A3| \u226bq 2d+1 3 or |A1|, |A2| \u226bq 3d+1 4 and |A3| \u226bq d+1 2 . More details will be provided in Section 5. The last theorems of this paper are on a Furstenberg type problem motivated by bounding the set in (1). We study the following question. Question 1.8. Let R be a set of rigid motions in Fd q and A \u2282Fd q, what can we say about the lower bound of the set S r\u2208R \u03c6r(A)? Since the size of S r\u2208R \u03c6r(A) is always bounded from below by |A|, we are interested in improving this trivial lower bound. Our \ufb01rst result reads as follows. Theorem 1.9. Let A be a set of points and R be a set of rigid motions in Fd q. We have \f \f \f \f \f [ r\u2208R \u03c6r(A) \f \f \f \f \f \u226bmin \u001a qd, |A||R| qd|O(d \u22121)| \u001b . In order to have the RHS of the above inequality larger than |A|, we need the condition that |R| \u226bqd|O(d \u22121)|. We emphasize here that this condition does not depend on the size of A, and is sharp if no additional assumption on A is added. To see this, let X \u2282Fq, we view O(d \u22121) as a subset of O(d) and de\ufb01ne Z := {(0, . . . , 0, \u2217): \u2217\u2208X}. Set A = (Fd\u22121 q \u00d7 {0}) + Z and R = O(d \u22121) \u00d7 A. It is clear that |R| = |O(d \u22121)||X|qd\u22121 and [ r\u2208R \u03c6r(A) \u2282Fd\u22121 q \u00d7 (X + X). Thus, given 0 < \u01eb < 1, if we take X to be an arithmetic progression in Fq of length q1\u2212\u01eb, then |R| = |O(d \u22121)|qd\u2212\u01eb and | [ r\u2208R \u03c6r(A)| \u22642|A|. In the next theorems, we provide improvements in which either the size of A is small or there is a relation between the sizes of A and R. Theorem 1.10. Let A be a set of points and R be a set of rigid motions in Fd q. Assume in addition that either (d \u22653 odd) or (d \u22612 mod 4 and q \u22613 mod 4). (1) If |A| < q d\u22121 2 , then we have \f \f \f \f \f [ r\u2208R \u03c6r(A) \f \f \f \f \f \u226bmin \u001a qd, |A||R| qd\u22121|O(d \u22121)| \u001b . 4 \f(2) If q d\u22121 2 \u2264|A| \u2264q d+1 2 , then we have \f \f \f \f \f [ r\u2208R \u03c6r(A) \f \f \f \f \f \u226bmin ( qd, |R| q d\u22121 2 |O(d \u22121)| ) . Theorem 1.11. Let A be a set of points and R be a set of rigid motions in F2 q with q \u22613 mod 4. We have \f \f \f \f \f [ r\u2208R \u03c6r(A) \f \f \f \f \f \u226bmin ( q2, |A|1/2|R| q ) . When A and R are small sets and R is a subset of oriented rigid motions, it is possible to obtain a further improvement. Let SO(2, q) be the group of orthogonal matrices with determinant 1. Let T(2, q) be the group of translations in F2 q, and SF(2, q) be the group of positively oriented rigid motions in F2 q, i.e. the group of elements of the from t \u25e6g, where t \u2208T(2, q) and g \u2208SO(2, q). De\ufb01ne SF \u2032(2, q) = SF(2, q) \\ T(2, q). Theorem 1.12. Let A be a set of points in F2 q and R be a set of rigid motions in SF \u2032(2, q) with q \u22613 mod 4. Assume that 2|R|1/5 < |A| < |R|3/5, then we have \f \f \f \f \f [ r\u2208R \u03c6r(A) \f \f \f \f \f \u226b|R|3/5. While we use incidence theorems developed in [16] to prove Theorems 1.9, 1.10, and 1.11, the proof of Theorem 1.12 requires a new incidence theorem for small sets of points and rigid motions (Theorem 2.8). This incidence theorem will be proved by using a point-line incidence bound in F3 q due to Koll\u00b4 ar [10]. We do not have any answer to the question on the sharpness of Theorems 1.10, 1.11, and 1.12, so we leave it as an open problem. We conclude this paper with some discussions on the case of prime \ufb01elds. 2 Incidences between points and rigid motions In this section, we recall incidence theorems between points and rigid motions in Fd q from [16] and state a new incidence theorem for small sets in F2 q. Let P = U \u00d7 V \u2282Fd q \u00d7 Fd q and R be a subset of O(d) \u00d7 Fd q. We say the point (u, v) is incident to the motion r = (g, z) if \u03c6r(u) := gu + z is equal to v. We denote the number of incidences between P and R by I(P, R). Theorem 2.1 ([16]). Let P \u2282F2d q and R be a subset of rigid motions in Fd q. Then the number of incidences between P and R satis\ufb01es I(P, R) \u2264|P||R| qd + Cqd/2p |O(d \u22121)| p |P||R|, for some large positive constant C. Corollary 2.2. Let P = U \u00d7 U \u2282Fd q \u00d7 Fd q. For k > 2|P|q\u2212d, let Rk be the number of elements of R incident to at least k elements from P. Then we have |Rk| \u226a|O(d \u22121)||P|qd k2 . 5 \fProof. We have k|Rk| \u2264I(P, Rk) \u2264|P||Rk| qd + Cqd/2p |O(d \u22121)| p |P||Rk|. Solving this inequality gives us the desired bound. If we put more conditions on d and q, the next two theorems present improvements. Theorem 2.3 ([16]). Let P = U \u00d7V for U, V \u2282Fd q. Assume in addition that either (d \u22653 odd) or (d \u22612 mod 4 and q \u22613 mod 4). There exists a large positive constant C such that the following hold. (1) If |U| < q d\u22121 2 , then we have I(P, R) \u2264|P||R| qd + Cq d\u22121 2 p |O(d \u22121)||P|1/2|R|1/2. (2) If q d\u22121 2 \u2264|U| \u2264q d+1 2 , then we have I(P, R) \u2264|P||R| qd + Cq d\u22121 4 p |O(d \u22121)||P|1/2|R|1/2|U|1/2. Theorem 2.4 ([16]). Let P = U \u00d7 V \u2282F2 q \u00d7 F2 q, |U| \u2264|V |, and R be a subset of O(2) \u00d7 F2 q with q \u22613 mod 4. Then we have I(P, R) \u2264|P||R| q2 + Cq1/2|U|3/4|V |1/2|R|1/2, for some large positive constant C. Corollary 2.5. Let P = U \u00d7 U \u2282F2 q \u00d7 F2 q with q \u22613 mod 4. For k > 2|P|q\u22122, let Rk be the number of elements of R incident to at least k elements from P. Then we have |Rk| \u226aq|P|5/4 k2 . Proof. The proof is the same as that of Corollary 2.2, namely, k|Rk| \u2264I(P, Rk) \u2264|P||Rk| q2 + Cq1/2|U|5/4|Rk|1/2. Solving this inequality gives us the desired bound. On the sharpness, it has been mentioned in [16] that Theorems 2.1 and 2.3 are sharp in odd dimensions. In terms of applications (Theorem 1.2), Theorem 2.4 is sharp in sense that the term q 1 2 |U|3/4|V |1/2|R| 1 2 cannot be improved to q 1 2 \u2212\u01eb|U|3/4|V |1/2|R| 1 2 for any \u01eb > 0. As proved in [16, Section 3] that one can use the Cauchy-Schwarz inequality to have I(P, R) \u2264|P||R|1/2 + |R|. (2) Assume |P| = |R| = N, so these two incidence bounds o\ufb00er the upper bound of N 3/2. Note that when N is small enough, this is better than that of Theorem 2.4 which depends on q. When q is 6 \fa prime number and P = U \u00d7 U with |U| \u226aq, it has been proved in [16, Section 3] that I(P, R) \u226a|P|5/6|R|1/2 + |R|. (3) In particular, if |P| = |R| = N, then I(P, R) \u226aN 4/3. If we put (2) and (3) together, the following questions appear naturally: Question 2.6. Suppose |P| = |R| = N, could we get an upper bound of the form N 3 2\u2212\u01eb, for some \u01eb > 0, over arbitrary \ufb01nite \ufb01elds? Question 2.7. Is it possible to have an upper bound of the form I(P, R) \u226a|P||R| 1 2 \u2212\u01eb for some \u01eb > 0? In the next theorem, we address these two questions for oriented rigid motions. Theorem 2.8. Let P = U \u00d7 U \u2282F4 q and R be a subset of SF \u2032(2, q) with q \u22613 mod 4. The number of incidences between P and R, denoted by I(P, R), satis\ufb01es I(P, R) \u226a|P||R|2/5 + |R|6/5. In particular, if |P| = |R| = N, then we have I(P, R) \u226aN 3 2 \u22121 10 . We remark here that our argument in the proof of Theorem 2.8 also works for other sets P satisfying the condition that min{|\u03c012(P|), |\u03c034(P)|} \u226a|P|1/2, where \u03c0ij(P) is the projection of P onto the two coordinates i and j. In the above theorem, the term |P||R|2/5 cannot be decreased to lower than |P||R|1/3. The reason is that one can take q = p3 with p \u22613 mod 4, P = U \u00d7 U \u2282F2 p \u00d7 F2 p with |P| = p3, and R = O(2, p) \u00d7 F2 p. Here O(2, p) is the set of orthogonal matrices with entries in Fp. Note that q = pr is 3 mod 4 if and only if p \u22613 mod 4 and r is odd. Thus, q = p3 with p \u22613 mod 4 satis\ufb01es the property q \u22613 mod 4. We can see that each point in P is incident to about |O(2, p)| = p elements of R. So, I(P, R) \u223c|P||R|1/3. 3 Triangles with one \ufb01xed side-length (Theorem 1.2) To prove Theorem 1.2, we recall the following result due to Shparlinski [17] on the number of \u201cunit\u201d distances in a pair of given sets in Fd q. When the two sets are the same, this was proved by Iosevich and Rudnev in [9]. Theorem 3.1. Let A, B \u2282F2 q. For \u03bb \u2208F\u2217 q, let N(\u03bb) be the number of pairs (x, y) \u2208A \u00d7 B such that ||x \u2212y|| = \u03bb. Then we have |A||B| q \u22124q 1 2 p |A||B| \u2264N(\u03bb) \u2264|A||B| q + 4q 1 2 p |A||B|. Let (x, y) be a line segment of length ||x \u2212y|| = \u03bb1 \u0338= 0 in F2 q. For \u03bb2, \u03bb3 \u2208Fq, it has been proved in [3] that this segment can be extended into at most two triangles (x, y, z) with ||x \u2212z|| = \u03bb2 and 7 \f||y \u2212z|| = \u03bb3. The precise statement is as follows. Lemma 3.2. Let (x, y)be a line segment of length ||x \u2212y|| = \u03bb1 \u0338= 0 in F2 q. For \u03bb2, \u03bb3 \u2208Fq, this segment can be extended into exactly \u00b5 triangles (x, y, z) with ||x \u2212z|| = \u03bb2 and ||y \u2212z|| = \u03bb3, where \u00b5 = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 2 if 4\u03c32 \u2212\u03c32 1 is a non-zero square in Fq 1 if 4\u03c32 \u2212\u03c32 1 is zero 0 if 4\u03c32 \u2212\u03c32 1 is a non-square in Fq and \u03c31 = \u03bb1 + \u03bb2 + \u03bb3, \u03c32 = \u03bb1\u03bb2 + \u03bb2\u03bb3 + \u03bb3\u03bb1. With these results, we are ready to prove Theorem 1.2. Proof of Theorem 1.2. Given \u03bb \u0338= 0, we know from Theorem 3.1 that the number of pairs (x, y) \u2208 A \u00d7 B such that ||x \u2212y|| = \u03bb is about \u223c|A||B| q . Let (x0, y0) be one of those pairs, for any pair (x, y) \u2208A\u00d7B such that (x, y) \u0338= (x0, y0) and ||x\u2212y|| = \u03bb, there exists unique r = (g, z) \u2208O(2)\u00d7F2 q such that gx0 + z = x and gy0 + z = y. Let R be the set of all such (g, z) corresponding to all pairs (x, y) with ||x \u2212y|| = \u03bb in A \u00d7 B. So, |R| \u223c|A||B| q . For r = (g, z) \u2208O(2) \u00d7 F2 q, we de\ufb01ne the map \u03c6r (or \u03c6g,z to be precise) from F2 q to F2 q by \u03c6r(x) = gx + z. Set C\u2032 := S r\u2208R \u03c6\u22121 r (C). It is enough to prove that |C\u2032| \u226bmin ( q2, |A||B||C|1/2 q2 ) \u226bq2, (4) whenever |A||B||C|1/2 \u226bq4. To see why we have at least \u226bq2 distinct congruence classes of triangles with one side-length \u03bb, we observe that for each v \u2208C\u2032, there exists (x, y, u) \u2208A \u00d7 B \u00d7 C and (g, z) \u2208R such that \u03c6\u22121 g,z(u) = v, \u03c6g,z(x0) = x, and \u03c6g,z(y0) = y. This means that the two triangles with vertices (x0, y0, v) and (x, y, u) are in the same congruence class. Using Lemma 3.2, we conclude that the number of distinct congruence classes of triangles is at least \u226bq2. To prove (4), we observe that I(C \u00d7 C\u2032, R\u22121) = |R||C|. Applying Theorem 2.4 gives us |R||C| \u226a|C||C\u2032||R| q2 + q1/2|C|3/4|C\u2032|1/2|R|1/2. Solving this inequality infers (4). This completes the proof of the theorem. Sharpness: To see the sharpness of this theorem, we construct an example as follows. Let C = {(0, 0)} \u2282F2 q and 0 < \u01eb < 1. We know that the group of rotations SO(2, q) in F2 q with q \u22613 mod 4 is cyclic of order q + 1. Assume q = pr, r is an exponent of three, and large enough such that (q + 1)/(p + 1) \u223cq1\u2212\u01eb. We recall the fact that q \u22613 mod 4 if and only if p \u22613 mod 4 and r is odd. Let \u03b8 \u2208SO(2, q) be an element of order k \u223cq1\u2212\u01eb. Let X be the set of t \u2208Fq \\ {0, 1} such that if t \u2208X then \u2212t \u0338\u2208X. Let v \u2208F2 q with ||v|| = 1, de\ufb01ne A = {v, \u03b8v, \u00b7 \u00b7 \u00b7 , \u03b8k\u22121v} [ t\u2208X {tv, t\u03b8v, \u00b7 \u00b7 \u00b7 , t\u03b8k\u22121v}. 8 \fWe have |A| = (|X| + 1) \u00b7 k \u223cq2\u2212\u01eb. We write A1 = {v, \u03b8v, \u00b7 \u00b7 \u00b7 , \u03b8k\u22121v}, At = {tv, t\u03b8v, \u00b7 \u00b7 \u00b7 , t\u03b8k\u22121v}, t \u2208X. By a direct computation, one can check that the set of distances between A\u03bb and A\u03b2, \u03bb, \u03b2 \u2208X\u222a{1}, is at most q1\u2212\u01eb. This implies the number of congruence classes of triangles with one vertex in C and the other two vertices in A is at most q3\u2212\u01eb. In other words, we have proved that for 0 < \u01eb < 1, there exists q = q(\u01eb) large enough such that there exist A, B, C with |A||B||C|1/2 = q4\u22122\u01eb and the number of congruence classes of triangles with vertices in A \u00d7 B \u00d7 C is at most q3\u2212\u01eb. 4 Extension in higher dimensions for simplex (Theorem 1.4) The idea to prove Theorem 1.4 is the same as that of Theorem 1.2. We need a more general version of Theorem 3.1 for (k \u22121)-simplex due to Vinh [18, 19]. Theorem 4.1. Let A1, . . . , Ak \u2282Fd q with |Ai| \u226bq d\u22121 2 +k\u22121. Then for any given (k \u22121)-simplex with non-zero side-lengths \u2206k\u22121, the number of copies of \u2206k\u22121 in Qk i=1 Ai is \u223cq\u2212(k 2) \u00b7 Qk i=1 |Ai|. The following extends Lemma 3.2 to higher dimensions, a proof can be found in [1, Lemma 3]. Lemma 4.2. Suppose we have k spheres S1, . . . , Sk with centers a1, . . . , ak and non-zero radii such that the system {a2 \u2212a1, . . . , ak \u2212a1} is linear independent, then |S1 \u2229S2 \u2229\u00b7 \u00b7 \u00b7 \u2229Sk| \u22642qd\u2212k. We are ready to provide proof of Theorem 1.4. Its proof is the same as that of Theorem 1.2, except that Theorem 2.4 is replaced by Theorem 2.1. Proof of Theorem 1.4. Using Theorem 4.1, we know that there are about \u223cQk i=1 |Ai|q\u2212(k 2) copies of \u2206k\u22121 in A. We \ufb01x one of them, say \u22060 k\u22121. Then for each copy \u2206i k\u22121, there are at least |O(d \u2212k + 1)| motions (g, z) in O(d) \u00d7 Fd q such that \u03c6g,z(\u2206i k\u22121) = \u22060 k\u22121. This comes from the fact that for any copy, we always can \ufb01nd a motion with that property and the stabilizer of each simplex is of the size \u223c|O(d \u2212(k \u22121))|. Let R be the set of these motions. So we can assume |R| \u223c|O(d \u2212k + 1)| Qk i=1 |Ai|q\u2212(k 2). As in the case k = 2, by using Lemma 4.2, it is enough to show that A\u2032 k+1 := S r\u2208R \u03c6\u22121 r (Ak+1) is of the size at least \u226bqd. We have, by Theorem 2.1, |Ak+1||R| \u2264I(Ak+1 \u00d7 A\u2032 k+1, R\u22121) \u2264|Ak+1||A\u2032 k+1||R| qd + |O(d \u22121)|1/2qd/2q |Ak+1||A\u2032 k+1||R|. This implies |A\u2032 k+1| \u226bmin ( qd, Qk+1 i=1 |Ai||O(d \u2212k + 1)| qd|O(d \u22121)|q(k 2) ) \u226bqd, whenever Qk+1 i=1 |Ai| \u226bqdk+1. 5 Rich rigid motions and simplices (Theorems 1.5 and 1.6) In this section, we prove Theorems 1.5 and 1.6. 9 \fLet P = A \u00d7 A \u2282Fd q \u00d7 Fd q and R be a set of rigid motions. For r = (g, z) \u2208O(d) \u00d7 Fd q, let i(r) be the number of pairs (x, y) \u2208P such that gx + z = y. Let Rk be the set of elements of R incident to at least k elements from P. Using upper bounds of Rk given in Corollaries 2.2 and 2.5, in the following propositions, we bound the sum P r i(r)t, for t \u22653, from above, where the sum runs over all rigid motions in O(d) \u00d7 Fd q. Proposition 5.1. Let P = A \u00d7 A with A \u2282Fd q. For r = (g, z) \u2208O(d) \u00d7 Fd q, let i(r) be the number of pairs (x, y) \u2208P such that gx + z = y. Then, for t \u22653, we have X r i(r)t \u226a|P|t|O(d)| q(t\u22121)d + qd|O(d \u22121)||P|t/2. Proof. Let Rk be the set of elements of R incident to at least k elements from P. Assume k > 2|P|q\u2212d, then, we know from Corollary 2.2 that |Rk| \u226a|O(d \u22121)||P|qd k2 . Using the fact that i(r) \u2264|A| for all r \u2208O(d) \u00d7 Fd q, we have X r\u2208O(d)\u00d7Fd q i(r)t = X r,i(r)<2|P |q\u2212d i(r)t + X r,i(r)>2|P |q\u2212d i(r)t \u226a \u0012|P| qd \u0013t \u00b7 |O(d)| \u00b7 qd + qd|P||O(d \u22121)||A|t\u22122 \u226a|P|t|O(d)| q(t\u22121)d + qd|O(d \u22121)||P|t/2. Thus, the theorem follows. In two dimensions and q \u22613 mod 4, we have a stronger estimate. Proposition 5.2. Let P = A \u00d7 A with A \u2282F2 q and q \u22613 mod 4. For r = (g, z) \u2208O(2) \u00d7 F2 q, let i(r) be the number of pairs (x, y) \u2208P such that gx + z = y. Then, for t \u22652, we have X r i(r)t \u226a|P|t q2t\u22123 + q|P| 2t+1 4 . Proof. Let Rk be the set of elements of R incident to at least k elements from P. Assume k > 2|P|q\u22122, then we know from Corollary 2.5 that |Rk| \u226aq|P|5/4 k2 . Using the fact that i(r) \u2264|A| for all r \u2208O(2) \u00d7 F2 q, one has X r\u2208O(2)\u00d7F2 q i(r)t = X r,i(r)<2|P |q\u22122 i(r)t + X r,i(r)>2|P |q\u22122 i(r)t \u2264|O(2)| \u00b7 q2 \u00b7 |P|t q2t + q|P|5/4|A|t\u22122 \u226a|P|t q2t\u22123 + q|P| 5 4+ t\u22122 2 = |P|t q2t\u22123 + q|P| 2t+1 4 . 10 \fThis completes the proof. We are now ready to prove Theorems 1.5 and 1.6. Proof of Theorem 1.5. We \ufb01rst start with the case k \u2264d. We recall that two k-simplices x = (x1, . . . , xk+1) and y = (y1, . . . , yk+1) are in the same congruence class if there exist g \u2208O(d) and z \u2208Fd q such that gxi + z = yi. We denote the set of equivalence classes by \u2206k(A). For each C \u2208\u2206k(A), by |C| we mean the number of k-simplices in that class. We \ufb01rst observe that |A|k+1 = X C\u2208\u2206k(A) |C|. Thus, by the Cauchy-Schwarz inequality, we have |A|2(k+1) \u2264|\u2206k(A)| \u00b7 # n (x, y) \u2208A2k+2 : x \u223cy o . In the next step, we need to bound the number of pairs (x, y) \u2208A2k+2 such that x \u223cy. For each k-simplex x = (x1, . . . , xk+1), by dim(x) we mean the dimension of the space spanned by vectors {x2 \u2212x1, . . . , xk+1 \u2212x1}. For each C \u2208\u2206k(A), we denote the stabilizer size of simplices in C by s(C). If dim(x) = \u03b3 and x \u2208C, then we know that s(C) \u223c|O(d \u2212\u03b3)|, see [4] for instance. Thus, for all C \u2208\u2206k(A), one has s(C) \u2265|O(d \u2212k)|. For each r = (g, z) \u2208O(d)\u00d7Fd q, as in Proposition 5.1, let i(r) be the number of pairs (u, v) \u2208A\u00d7A such that gu + z = v. It is clear that # n (x, y) \u2208A2k+2 : x \u223cy o \u2272 1 |O(d \u2212k)| X r\u2208O(d)\u00d7Fd q i(r)k+1 \u2272 1 |O(d \u2212k)| \u00b7 \u0012|P|k+1|O(d)| qdk + qd|O(d \u22121)||P|(k+1)/2 \u0013 \u2272|A|2k+2|O(d)| q(d\u2212k 2 )+dk + qd+(d\u22121 2 )\u2212(d\u2212k 2 )|A|k+1 \u226a|A|2k+2 q(k+1 2 ) , whenever |A| \u226bq dk+1 k+1 . In other words, under this condition, the number of congruence classes of k-simplex in A is at least \u226bq(k+1 2 ). On the other hand, it has been proved in [4] that, the number of congruence classes of degenerate k-simplex in Fd q is at most o \u0010 q(k+1 2 )\u0011 . We now move to the case k > d. In this case, we say a k-simplex x = (x1, . . . , xk+1) is nondegenerate if dim(x) = d, and degenerate otherwise. It has been proved in [14] that for k > d, the number of congruence classes of k-simplex in Fd q in total is \u223cqd(k+1)\u2212(d+1 2 ). It was also proved in [14] that the number of degenerate k-simplex is o \u0010 qd(k+1)\u2212(d+1 2 )\u0011 . 11 \fWhen k > d, we proceed as above, the only di\ufb00erence here is that # n (x, y) \u2208E2k+2 : x \u223cy o \u2272 X r\u2208O(d)\u00d7Fd q i(r)k+1 \u2272|P|k+1|O(d)| qdk + qd|O(d \u22121)||P|(k+1)/2 \u2272|A|2k+2|O(d)| qdk + qd+(d\u22121 2 )\u2212(d\u2212k 2 )|A|k+1 \u226a |A|2k+2 qd(k+1)\u2212(d+1 2 ) , whenever |A| \u226bq dk+1 k+1 . This completes the proof of the theorem. Proof of Theorem 1.6. The proof of Theorem 1.6 is the same as that of dimension d, except that we use Proposition 5.2 in place of Proposition 5.1. More precisely, we will have # n (x, y) \u2208A2k+2 : x \u223cy o \u2272 X r\u2208O(2)\u00d7F2 q i(r)k+1 \u226a|A|2k+2 q2k\u22121 + q|A| 2k+3 2 \u226a|A|2k+2 q2k\u22121 , whenever |A| \u226bq 4k 2k+1 . In conclusion, under |A| \u226bq 4k 2k+1, we have |\u2206k(A)| \u226bq2k\u22121. This completes the proof of the theorem. We conclude this section with some discussions on the case of di\ufb00erent underlying sets. For the sake of simplicity, let us consider the case k = 2 in Fd q. Let A, B, C be sets in Fd q, to prove that A \u00d7 B \u00d7 C determines a positive proportion of all congruence classes of triangles, we need to show that # \b (x, y) \u2208(A \u00d7 B \u00d7 C)2 : x \u223cy \t \u2272 1 |O(d \u22122)| X r\u2208O(d)\u00d7Fd q iA(r)iB(r)iC(r) \u2272 1 |O(d \u22122)| \u00b7 \u0012|A|2|B|2|C|2|O(d)| q2d \u0013 , here iU(r) is the number of pairs (x, y) \u2208U \u00d7 U such that \u03c6r(x) = gx + z = y. By the H\u00a8 older inequality, we have X r\u2208O(d)\u00d7Fd q iA(r)iB(r)iC(r) \u2264  X r iA(r)3 !1/3 \u00b7  X r iB(r)3 !1/3 \u00b7  X r iC(r)3 !1/3 . Applying Proposition 5.1, we need the conditions that |A|, |B|, |C| \u226bq 2d+1 3 . If we use di\ufb00erent exponents in the H\u00a8 older step, namely, X r\u2208O(d)\u00d7Fd q iA(r)iB(r)iC(r) \u2264  X r iA(r)4 !1/4 \u00b7  X r iB(r)4 !1/4 \u00b7  X r iC(r)2 !1/2 , 12 \fthen we will need |A|, |B| \u226bq 3d+1 4 , |C| \u226bq d+1 2 . In other words, compared to these results, Theorem 1.4 o\ufb00ers the best conditions in practice. 6 Furstenberg type problem for rigid motions (Theorems 1.9, 1.10, 1.11, and 1.12) The proofs of Theorems 1.9, 1.10, 1.11, and 1.12 are the same. Thus, we only present that of Theorem 1.9. More precisely, we set B = S r\u2208R \u03c6r(A), then it is clear that I(A \u00d7 B, R) = |R||A|. Applying Theorem 2.1, one has I(A \u00d7 B, R) \u2264|A||B||R| qd + Cq d 2 p |O(d \u22121)||A|1/2|B|1/2|R|1/2. Putting lower and upper bounds together, we get the desired result. We note here that in the proofs of Theorems 1.10-1.12, we use Theorems 2.3, 2.4, and 2.8, respectively. 7 Incidences for small sets (Theorem 2.8) Let p1, p2, p3, p4 \u2208F2 q be points such that ||p1\u2212p3|| = ||p2\u2212p4|| \u0338= 0 and p1\u2212p3 \u0338= p2\u2212p4. We know that there exists a unique pair (g, z) with g \u2208SO(2, q) \\ {I} and z \u2208F2 q such that gp1 + z = p2 and gp3 + z = p4. Since q \u22613 mod 4, it has been indicated in [3] that there is a natural way to identify the group SO(2, q) and the \ufb01eld Fq. More precisely, we consider the map \u03d5: Fq \u2192SO(2, q) \\ {I} de\ufb01ned by \u03d5(r) =   r2\u22121 r2+1 \u22122r r2+1 2r r2+1 r2\u22121 r2+1 ! . Note that r2 + 1 \u0338= 0 under q \u22613 mod 4. From now we identify each matrix in SO(2, q) \\ {I} with the corresponding element in F2 q. Given p, q \u2208R2, it is well-known that the set of rigid motions in R2 that map p to q can be parameterized as a line in R3, see [5, 7] or [8, Chapter 9] for instance. Thus, the number of incidences between points and rigid motions in R2 can be reduced to the number of incidences between points and lines in R3. To prove Theorem 2.8, we will apply the same strategy. We \ufb01rst recall the following lemma from [3]. Lemma 7.1 ([3]). Given p, q \u2208F2 q. De\ufb01ne \u2113p\u2192q be the set of points \u0000(I \u2212g)\u22121z, r \u0001 , where gp+z = q, g = \u03d5(r) \u2208SO(2, q)\\{I}, z \u2208F2 q, then \u2113p\u2192q is a line in F3 q. In particular, \u2113p\u2192q can be presented in the following form \u001a\u0012p + q 1 , 0 \u0013 + r \u0012(p \u2212q)\u22a5 2 , 1 \u0013 : r \u2208Fq \u001b . Here (p1, p2)\u22a5= (p2, \u2212p1). 13 \fIn F3 q, the following point-line incidence bound is due to Koll\u00b4 ar in [10]. Theorem 7.2 ([10]). Let P be a set of n distinct points and L be a set of m distinct lines in F3 q. Assume that no plane contains more than c\u221am lines from L for some constant c. Then we have I(P, L) \u226a|L||P|2/5 + |P|6/5. Using Lemma 7.1 above, one can see that the number of incidences between P and R can be reduced to the number of incidences between points and lines in F3 q. Let P and L be the set of corresponding points and lines in F3 q. So we can apply Theorem 7.2 as long as we know that any plane contains only few lines. The rest of this section is devoted to proving such a property. We \ufb01rst observe that the number of lines is equal to |P| = |U|2. This means that we need to show that each plane contains at most \u226a|U| lines. Given a plane \u03c0, it is clear that any family of parallel lines contains at most |U| lines, and the same happens for any pencil, i.e. families of concurrent lines. Lemma 7.3. If there exist three lines on \u03c0 \u2229L that are non-concurrent and pairwise intersecting, then \u03c0 contains at most |U| lines from L. Proof. Assume those three lines are \u2113pi\u2192qi, 1 \u2264i \u22643, then we can check that the two triangles (p1, p2, p3) and (q1, q2, q3) are in the same congruence class under a rigid motion with an orthogonal matrix of determinant \u22121. Let g =   a b b \u2212a ! , a2 + b2 = 1, be an orthogonal matrix with det(g) = \u22121, and z := (z1, z2) \u2208F2 q. For x = (x1, x2) \u2208F2 q, the map \u03c6g,z is de\ufb01ned by \u03c6g,z(x1, x2) =   a b b \u2212a ! \u00b7   x1 x2 ! +   z1 z2 ! =   ax1 + bx2 + z1 bx1 \u2212ax2 + z2 ! . Under this map,we have (x1, x2) \u2192(ax1 + bx2 + z1, bx1 \u2212ax2 + z2), (y1, y2) \u2192(ay1 + by2 + z1, by1 \u2212ay2 + z2). By abuse of notation, let \u2113(x1,x2),\u03c6g,z be the line de\ufb01ned as in Lemma 7.1 by two points (x1, x2) and \u03c6g,z(x1, x2). Then the line \u2113(x1,x2),\u03c6g,z contains points of the form \u0012x1(a + 1) + bx2 + z1 2 , bx1 + x2(1 \u2212a) + z2 2 , 0 \u0013 +r \u0012x2(1 + a) \u2212bx1 \u2212z2 2 , (a \u22121)x1 + bx2 + z1 2 , 1 \u0013 , and the line \u2113(y1,y2),\u03c6g,z contains points of the form \u0012y1(a + 1) + by2 + z1 2 , by1 + y2(1 \u2212a) + z2 2 , 0 \u0013 +r \u0012y2(1 + a) \u2212by1 \u2212z2 2 , (a \u22121)y1 + by2 + z1 2 , 1 \u0013 , where r \u2208Fq. Assume these two lines span a plane denoted by \u03c0. 14 \fWe now show that given (u1, u2), (v1, v2) \u2208F2 q, if the line containing points of the form \u0012u1 + v1 2 , u2 + v2 2 , 0 \u0013 + r \u0012u2 \u2212v2 2 , v1 \u2212u1 2 , 1 \u0013 . lies on \u03c0, then we will have   a b b \u2212a ! \u00b7   u1 u2 ! +   z1 z2 ! =   v1 v2 ! . This implies that the plane \u03c0 contains at most |U| lines as desired and we are done. Since any plane is of dimension two, there exist (m, n) \u0338= (0, 0) and m + n = 1 such that u2 \u2212v2 = m ((1 + a)x2 \u2212bx1 \u2212z2) + n ((1 + a)y2 \u2212by1 \u2212z2) , and v1 \u2212u1 = m((a \u22121)x1 + bx2 + z1) + n((a \u22121)y1 + by2 + z1). We then can write v1 = u1 + (a \u22121)(mx1 + ny1) + b(mx2 + ny2) + z1 = au1 + bu2 + (a \u22121)(mx1 + ny1 \u2212u1) + b(mx2 + ny2 \u2212u2) + z1, and v2 = u2 \u2212m((1 + a)x2 \u2212bx1 \u2212z2) \u2212n((1 + a)y2 \u2212by1 \u2212z2) = bu1 \u2212au2 \u2212(1 + a)(mx2 + ny2 \u2212u2) + b(mx1 + ny1 \u2212u1) + z2. It su\ufb03ces to prove that (a\u22121)(mx1 +ny1 \u2212u1)+b(mx2 +ny2 \u2212u2) = 0, (1+a)(mx2 +ny2 \u2212u2)+b(mx1 +ny1 \u2212u1) = 0. Note that the plane \u03c0 intersects the plane Z = 0 in F3 q at the line denoted by \u2113(Z = 0) de\ufb01ned by \u001a\u0012(a + 1)x + by + z1 2 , bx + y(1 \u2212a) + z2 2 \u0013 : (x, y) \u2208F2 q \u001b . To simplify the notations, we assume that (z1, z2) = (0, 0). Note that the point ((u1 + v1)/2, (u2 + v2)/2) lies on this line. This point can be rewritten as =: \u0012u1(a + 1) + bu2 2 , bu1 \u2212(a \u22121)u2 2 \u0013 + (A/2, B/2), where (A, B) := ((a\u22121)(mx1+ny1\u2212u1)+b(mx2+ny2\u2212u2), \u2212(a+1)(mx2+ny2\u2212u2)+b(mx1+ny1\u2212u1)). Note that \u0010 u1(a+1)+bu2 2 , bu1\u2212(a\u22121)u2 2 \u0011 \u2208\u2113(Z = 0) by a direct computation. If (u1(a + 1) + bu2, bu1 \u2212(a \u22121)u2) + (A, B) \u2208\u2113Z=0, 15 \fthen the direction of \u2113(Z = 0) is parallel to (A, B) in the plane Z = 0. On the other hand, (0, 0) \u2208\u2113(Z = 0) and 1 2 \u00b7   (a + 1) b b (1 \u2212a) ! \u00b7   \u2212mx2 \u2212ny2 + u2 mx1 + ny1 \u2212u1 ! = (B/2, \u2212A/2) \u2208\u2113(Z = 0). This means that (A, B) = \u03bb(B, \u2212A) for some \u03bb \u0338= 0. This infers A = B = 0. In other words, any line of the form \u0012u1 + v1 2 , u2 + v2 2 , 0 \u0013 + r \u0012u2 \u2212v2 2 , v1 \u2212u1 2 , 1 \u0013 belonging to \u03c0 will satisfy the relation   a b b \u2212a ! \u00b7   u1 u2 ! +   z1 z2 ! =   v1 v2 ! . This completes the proof. To conclude that the intersection \u03c0 \u2229L contains at most \u226a|U| lines, we decompose \u03c0 \u2229L into families of parallel lines, say, F1, F2, . . . , Fm. If m \u22642, we are done. Thus, we can assume that m \u22653. If there are three lines satisfying the above lemma, we are also done. Thus, we can assume that \u03c0 \u2229L contains no three such lines. Let \u21131 \u2208F1 and \u21132 \u2208F2. Set p = \u21131 \u2229\u21132, then all lines in Fi, i \u22653, have to pass through p. This means that |Fi| \u22641 for all i \u22653. Using the fact that for each p, there are at most |U| lines containing it, so in total, the intersection \u03c0 \u2229L contains at most \u226a|U| lines. 8 Discussions over prime \ufb01elds We might wonder if the recent progress on the distance problem over prime \ufb01elds due to Murphy, Petridis, Pham, Rudnev, and Stevens [15] is helpful to study the simplex problem. In this section, we discuss this direction. We adapt the method in Section 5 to prove the following theorem. Theorem 8.1. Let A, B, C \u2282F2 p with p 5 4 \u2264|C| \u2264p 4 3 and p \u22613 mod 4. Assume that |A|, |B| \u226b p12/7, then the number of congruence classes of triangles with vertices in A \u00d7 B \u00d7 C is at least \u226bp3. Remark 8.1. Compared to Theorem 1.2, this result is much weaker. In Theorem 8.1, we only assumed that |C| \u2208(p5/4, p4/3), if the same condition applies for A or B, then the condition will become worse. When A = B = C (for simplicity) and |A| = |B| = |C| < p5/4, it is not di\ufb03cult to see that there will be at least |\u2206(A)| \u00b7 |A| congruence classes with vertices in A \u00d7 B \u00d7 C. Here \u2206(A) is the set of distinct distances determined by pairs of points in A. We now prove Theorem 8.1. In the plane over prime \ufb01elds, the following incidence theorem was proved in [16]. Theorem 8.2 ([16]). Let P = U \u00d7U \u2282F2 p \u00d7F2 p and R be a subset of O(2)\u00d7F2 p with p \u22613 mod 4. Assume that p5/4 \u2264|U| \u2264p4/3, then we have I(P, R) \u2264|P||R| p2 + Cp1/8|U|3/2|R|1/2, 16 \ffor some large positive constant C. With the same argument as in arbitrary \ufb01nite \ufb01elds, we have the following corollary. Corollary 8.3. Let P = U \u00d7 U \u2282F2 p \u00d7 F2 p with p \u22613 mod 4 and p5/4 \u2264|U| \u2264p4/3. For k > 2|P|p\u22122, let Rk be the number of elements of R incident to at least k elements from P. Then we have |Rk| \u226ap1/4|U|3 k2 . For r = (g, z) \u2208O(2) \u00d7 F2 p, let iU(r) be the number of pairs (x, y) \u2208U \u00d7 U such that gx + z = y. Then, for t \u22652, we have X r i(r)t \u226a|P|t p2t\u22123 + p1/4|U|t+1. Proof of Theorem 8.1. As in the case of Fq, we have # \b (x, y) \u2208(A \u00d7 B \u00d7 C)2 : x \u223cy \t \u2272 X r\u2208O(2)\u00d7F2 q iA(r)iB(r)iC(r) \u2264  X r iA(r)4 !1/4 \u00b7  X r iB(r)4 !1/4 \u00b7  X r iC(r)2 !1/2 . It is su\ufb03cient to show that # \b (x, y) \u2208(A \u00d7 B \u00d7 C)2 : x \u223cy \t \u226a|A|2|B|2|C|2 p3 when |A|, |B| \u226bp12/7. Using Proposition 5.2, we have X r iA(r)4 \u226a|A|8 p5 + p|A|9/2, and X r iB(r)4 \u226a|B|8 p5 + p|B|9/2. The above corollary gives X r iC(r)2 \u226a|C|4 p . In total, # \b (x, y) \u2208(A \u00d7 B \u00d7 C)2 : x \u223cy \t \u226a|A|2|B|2|C|2 p3 +|A|9/8|B|9/8|C|2+|A|2|B|9/8|C|2 p3/2 +|B|2|A|9/8|C|2 p3/2 . This is at most \u226a|A|2|B|2|C|2/p3 whenever |A|, |B| \u226bp12/7. This completes the proof of the theorem. 17 \fWe want to add a remark here that if we use the bound X r\u2208O(2)\u00d7F2 q iA(r)iB(r)iC(r) \u2264  X r iA(r)3 !1/3 \u00b7  X r iB(r)3 !1/3 \u00b7  X r iC(r)3 !1/3 then we will never get a positive proportion of all congruence classes of triangles no matter how large A and B are. This happens when the size of C is smaller than p3/2. 9 Acknowledgements T. Pham would like to thank the Vietnam Institute for Advanced Study in Mathematics (VIASM) for the hospitality and for the excellent working condition."
+                },
+                {
+                    "url": "http://arxiv.org/abs/2304.09464v1",
+                    "title": "A discretized point-hyperplane incidence bound in $\\mathbb{R}^d$",
+                    "abstract": "Let $P$ be a $\\delta$-separated $(\\delta, s, C_P)$-set of points in $B(0,\n1)\\subset \\mathbb{R}^d$ and $\\Pi$ be a $\\delta$-separated $(\\delta, t,\nC_\\Pi)$-set of hyperplanes intersecting $B(0, 1)$ in $\\mathbb{R}^d$. Define\n  \\[I_{C\\delta}(P, \\Pi)=\\#\\{(p, \\pi)\\in P\\times \\Pi\\colon p\\in\n\\pi(C\\delta)\\}.\\] Suppose that $s, t\\ge \\frac{d+1}{2}$, then we have\n$I_{C\\delta}(P, \\Pi)\\lesssim \\delta |P||\\Pi|$. The main ingredient in our\nargument is a measure theoretic result due to Eswarathansan, Iosevich, and\nTaylor (2011) which was proved by using Sobolev bounds for generalized Radon\ntransforms. Our result is essentially sharp, a construction will be provided\nand discussed in the last section.",
+                    "authors": "Thang Pham, Chun-Yen Shen, Nguyen Pham Minh Tri",
+                    "published": "2023-04-19",
+                    "updated": "2023-04-19",
+                    "primary_cat": "math.CA",
+                    "cats": [
+                        "math.CA",
+                        "math.CO",
+                        "math.NT"
+                    ],
+                    "main_content": "Introduction We start with the following two de\ufb01nitions. De\ufb01nition 1.1 ((\u03b4, s, C) set). Let 0 \u2264s < \u221eand \u03b4 \u2208(0, 1) and a constant C > 0. Given a metric space (X, d), a bounded set E \u2282X is called (\u03b4, s, C)-set if for every \u03b4 \u2264r \u22641 and for all ball B \u2282X of radius r, we have |E \u2229B|\u03b4 \u2264Crs|E|\u03b4, where |E|\u03b4 denotes the \u03b4-covering of E in the space (X, d). We note that this de\ufb01nition is slightly di\ufb00erent compared to the classical one introduced by Katz and Tao in [9]. De\ufb01nition 1.2 (Katz-Tao (\u03b4, s, C)-set). Let 0 \u2264s < \u221eand \u03b4 \u2208(0, 1) and a constant C > 0. Given a metric space (X, d), a bounded set E \u2282X is called Katz-Tao (\u03b4, s, C)-set if for every \u03b4 \u2264r \u22641 and for all ball B \u2282X of radius r, we have |E \u2229B|\u03b4 \u2264C \u0010r \u03b4 \u0011s . Note that if |E|\u03b4 \u223c\u03b4\u2212s, then the two above de\ufb01nitions are equivalent. Let P be a \u03b4-separated (\u03b4, s, CP )-set of points in Rd and \u03a0 be a \u03b4-separated (\u03b4, t, C\u03a0)-set of hyperplanes intersecting B(0, 1) in Rd. The number of discretized point-plane incidences between P and \u03a0 is de\ufb01ned by IC\u03b4(P, \u03a0) = #{(p, \u03c0) \u2208P \u00d7 \u03a0: p \u2208\u03c0(C\u03b4)}, where \u03c0(C\u03b4) denotes the C\u03b4 neighborhood of the hyperplane \u03c0. In this paper, we treat CP , C\u03a0, and other constants as absolutely positive bounded constants. \u2217University of Science, Vietnam National University, Hanoi. Email: phamanhthang.vnu@gmail.com \u2020Department of Mathematics, National Taiwan University. Email: cyshen@math.ntu.edu.tw \u2021Ho Chi Minh University of Education. Email: nguyenphamminhtri104@gmail.com 1 \fOur initial motivation comes from the following recent theorem due to Orponen, Shmerkin, and Wang [11], and Fu and Ren [4] in two dimensions. Theorem 1.3. Let 0 \u2264s, t \u22642. Then, for every \u01eb > 0, there exists \u03b40 = \u03b40(\u01eb) > 0 such that the following holds for all \u03b4 \u2208(0, \u03b40]. Let P \u2282B(0, 1) \u2282R2 be a \u03b4-separated (\u03b4, s, \u03b4\u2212\u01eb)-set of points and T be a \u03b4-separated (\u03b4, t, \u03b4\u2212\u01eb)-set of tubes intersecting B(0, 1). 1. If 1 \u2265t \u2265s or 1 \u2265s \u2265t, then I\u03b4(P, T ) \u2272|P||T |\u03b4 st s+t\u2212O(\u01eb). 2. If t \u22651 \u2265s \u2265t \u22121, then I\u03b4(P, T ) \u2272|P||T |\u03b4 st 1+s\u2212O(\u01eb) 3. If s \u22651 \u2265t \u2265s \u22121, then I\u03b4(P, T ) \u2272|P||T |\u03b4 st 1+t \u2212O(\u01eb). 4. If t > 1 and s > 1, then I\u03b4(P, T ) \u2272|P||T |\u03b4\u03ba(s+t\u22121)\u2212O(\u01eb), \u03ba = min{1/2, 1/(s + t \u22121)}. Throughout this paper, by X \u2272Y we mean that X \u2264CY for some constant C, and X \u223cY if X \u2272Y \u2272X. The main purpose of this paper is to extend Theorem 1.3 to higher dimensions, namely, in Rd with d \u22653. Our \ufb01rst result is the following. Theorem 1.4. Let C > 0, 0 \u2264s, t \u2264d. There exists \u03b40 = \u03b40(C, s, t) > 0 such that the following holds for \u03b4 \u2208(0, \u03b40). Let P be a \u03b4-separated (\u03b4, s, CP )-set of points in B(0, 1) \u2282Rd and \u03a0 be a \u03b4-separated (\u03b4, t, C\u03a0)-set of hyperplanes intersecting B(0, 1) in Rd. De\ufb01ne IC\u03b4(P, \u03a0) = #{(p, \u03c0) \u2208P \u00d7 \u03a0: p \u2208\u03c0(C\u03b4)}. Suppose that s, t > d+1 2 , then we have the sharp estimate that IC\u03b4(P, \u03a0) \u2272\u03b4|P||\u03a0|. In the above theorem, the conditions s > d+1 2 and t > d+1 2 are required in the proof. When s, t are small, using an elementary geometric argument, we are able to prove the following non-trivial result. Theorem 1.5. Let C > 0, 0 \u2264s, t \u2264d. There exists \u03b40 = \u03b40(C, s, t) > 0 such that the following holds for \u03b4 \u2208(0, \u03b40). Let P be a (\u03b4, s, CP ) set of points in B(0, 1) \u2282Rd and \u03a0 be a (\u03b4, t, C\u03a0) set of hyperplanes intersecting B(0, 1) in Rd with d \u22653. We further assume that s \u2212d + 2 > 0. Then, for any \u01eb > 0, we have IC\u03b4(P, \u03a0) \u2272|P| \u00b7 |\u03a0| \u00b7 \u03b4f(t)(s\u2212d+2)\u2212\u01eb, where f(t) = \uf8f1 \uf8f2 \uf8f3 1/2, if t \u22651 t 1 + t, if t < 1 . Remark 1.1. If t \u2212d + 2 > 0, then we can use the dual arguments to obtain a similar result. While Theorem 1.4 is optimal, we do not have any constructions on the sharpness of this theorem. 2 \fMain ideas and Comparisons: We \ufb01rst discuss the main idea in the proof of Theorem 1.3. Observe that if P is a (\u03b4, s, C)-set, then P can be covered by at most |P|\u03b4s\u2212O(\u01eb) Katz-Tao (\u03b4, s)sets. A detailed proof can be found in [11, Lemma 3.5]. With this observation, one can apply Theorem 1.4 and Theorem 1.5 due to Fu and Ren in [4]. To prove these two theorems, Fu and Ren used a geometric argument and some earlier results due to Guth, Solomon, and Wang [5] and a generalization due to Bradshaw [1]. In higher dimensions d \u22653, to prove Theorem 1.4, we use a completely di\ufb00erent approach. More precisely, we use a measure theoretic result due to Eswarathansan, Iosevich, and Taylor [3], which was proved by using Sobolev bounds for generalized Radon transforms. The proof of Theorem 1.5 is a combination of Cauchy-Schwartz argument and geometric results due to Hera, Keleti, and Mathe [6]. We also want to add a remark that the approach in the proof of Theorem 1.4 is similar to the mechanism introduced by Iosevich, Jorati, and Laba [8] when they studied incidences between a set of \u201ctubes\u201d and a homogeneous set of points with the same size. Since our proof uses Sobolev bounds for generalized Radon transforms, it can be generalized to a more general form, i.e. the equation of hyperplanes xd = a1x1 + \u00b7 \u00b7 \u00b7 + ad\u22121xd\u22121 + ad can be replaced by \u03a8(a1, . . . , ad, x1, . . . , xd) = 0, where the function \u03a8 satis\ufb01es the Phong-Stein curvature condition (4) below. If the set \u03a0 of planes only needs to satisfy the property that it is \u03b4-separated, a recent work of Dabrowski, Orponen, and Villa [2] for the case of (d \u22121)-hyperplanes tells us that IC\u03b4(P, \u03a0) \u2272\u03b4\u2212\u01eb\u03b4 (d\u22121)(s+1\u2212d) 2d\u22121\u2212s |P||\u03a0| d\u22121 2d\u22121\u2212s , (1) where P is a \u03b4-separated (\u03b4, s, CP )-set with s > 1. This result is weaker than Theorem 1.4 when |\u03a0| \u2264\u03b4\u2212d. We now compare the estimate (1) and Theorem 1.5. Theorem 1.5 states that if s \u2212d + 2 > 0, then IC\u03b4(P, \u03a0) \u2272|P| \u00b7 |\u03a0| \u00b7 \u03b4f(t)(s\u2212d+2)\u2212\u01eb, where f(t) = \uf8f1 \uf8f2 \uf8f3 1/2, if t \u22651 t 1 + t, if t < 1 . Assume t \u22651, then by a direct computation, this bound is stronger than that of (1) when \u03b4\u2212t \u2272|\u03a0| \u2264\u03b4 M(2d\u22121\u2212s) d\u2212s , where M = (d \u22121)(s + 1 \u2212d) 2d \u22121 \u2212s \u2212s \u2212d + 2 2 . This range is non-empty when 2t \u2264s + 1 < d. Assume t < 1, then Theorem 1.5 is stronger than that of (1) when \u03b4\u2212t \u2272|\u03a0| \u2264\u03b4 M\u2032(2d\u22121\u2212s) d\u2212s , where M\u2032 = M \u2212 \u0012 t t + 1 \u22121 2 \u0013 (s \u2212d + 2). 3 \fThis range is non-empty when d \u22122 < s < d \u22121. 2 Proof of Theorem 1.4 We \ufb01rst recall some notations that can be found in [2]. Let A(d, n) be the set of n-dimensional a\ufb03ne subspaces in Rd. The metric dA de\ufb01ned on A(d, n) ([10, page 53]) is de\ufb01ned by dA(V, W) = ||\u03c0V0 \u2212\u03c0W0||op + |a \u2212b|, where V = V0 +a and W = W0 +b, V0, W0 \u2208G(d, n) (the set of n-dimensional subspaces), a \u2208V \u22a5 0 , b \u2208W \u22a5 0 , and || \u00b7 ||op is the operator norm. Let \u03b31 and \u03b32 be two (d \u22121)-planes de\ufb01ned by \u03b31 : xd = a1x1 + \u00b7 \u00b7 \u00b7 + ad\u22121xd1 + ad, and \u03b32 : xd = b1x1 + \u00b7 \u00b7 \u00b7 + bd\u22121xd1 + bd. The following formula is more useful in practise dA (\u03b31, \u03b32) = \f \f \f \f (a1 . . . , ad\u22121, \u22121) |(a1 . . . , ad\u22121, \u22121| \u2212(b1 . . . , bd\u22121, \u22121) |(b1 . . . , bd\u22121, \u22121| \f \f \f \f + \f \f \f \f ad |(a1 . . . , ad\u22121, \u22121| \u2212 bd |(b1 . . . , bd\u22121, \u22121)| \f \f \f \f . This can be proved by a direct computation or can be found in [7, Lemma 2.5]. For each plane de\ufb01ned by xd = a1x1 + \u00b7 \u00b7 \u00b7 + ad\u22121xd\u22121 + ad, we call the point (a1, . . . , ad) its dual point in Rd. Let D: Rd \u2192A(d, d \u22121) be the map de\ufb01ned by D: (x1, . . . , xd) 7\u2192 ( yd = d\u22121 X i=1 xiyi + xd ) . There are some properties of D we should keep in mind, namely, the restriction of D to the ball B(R), 0 < R < \u221e, is bilipschitz onto its image, and the bilipschitz constant depends only on R and d. Thus, D is injective. The inverse map of D, denoted by D\u2217: im(D) \u2192Rd, de\ufb01ned by D(x1, . . . , xd) 7\u2192(\u2212x1, . . . , \u2212xd\u22121, xd). There exists a dimensional constant 0 < rd < 1 such that the restriction of D\u2217to B(V0, rd) is bilipschitz onto its image, here V0 = D(0, . . . , 0). With these notations, the next two lemmas can be proved with standard arguments. Lemma 2.1. Let S be a set of \u03b4-separated hyperplanes intersecting the unit ball B(0, 1). Then the set of corresponding dual points is also c\u03b4-separated for some absolute constant c. Proof. If two hyperplanes \u03b31 and \u03b32 are \u03b4-separated with the metric dA (\u03b31, \u03b32) = \f \f \f \f (a1 . . . , ad\u22121, \u22121) |(a1 . . . , ad\u22121, \u22121| \u2212(b1 . . . , bd\u22121, \u22121) |(b1 . . . , bd\u22121, \u22121| \f \f \f \f + \f \f \f \f ad |(a1 . . . , ad\u22121, \u22121| \u2212 bd |(b1 . . . , bd\u22121, \u22121)| \f \f \f \f , 4 \fwhere \u03b31 and \u03b32 are respectively de\ufb01ned by equations (\u03b31) : yd = d\u22121 X i=1 aiyi + ad, and (\u03b32) : yd = d\u22121 X i=1 biyi + bd, then we want to prove that the usual Euclidean distance between two points (a1, . . . , ad) and (b1, . . . , bd) is at least c\u03b4 for some absolute constant c. For the sake of simplicity, we will denote u = (a1 . . . , ad\u22121, \u22121) and v = (b1 . . . , bd\u22121, \u22121). Then clearly |u|, |v| \u22651. Since two hyperplanes \u03b31, \u03b32 are \u03b4-separated, we have that dA (\u03b31, \u03b32) \u2265\u03b4. With these assumptions, we attempt to prove that v u u t d X i=1 (ai \u2212bi)2 \u2273\u03b4 It should be noted that, if we view ad, bd as vectors (0, . . . , ad), (0, . . . , bd) respectively, then the usual Euclidean distance between two points (a1, . . . , ad) and (b1, . . . , bd) can be rewritten as p |u \u2212v|2 + |ad \u2212bd|2. Thus, the goal is to prove the following inequality \f \f \f \f u |u| \u2212v |v| \f \f \f \f + \f \f \f \f ad |u| \u2212bd |v| \f \f \f \f \u2272 p |u \u2212v|2 + |ad \u2212bd|2. (2) Instead of proving 2, we prove a stronger version \f \f \f \f u |u| \u2212v |v| \f \f \f \f + \f \f \f \f ad |u| \u2212bd |v| \f \f \f \f \u2272|u \u2212v| + |ad \u2212bd|. (3) We \ufb01rst prove \f \f \f \f u |u| \u2212v |v| \f \f \f \f \u2264|u \u2212v|. Squaring both sides gives |u|2 + |v|2 \u22122u \u00b7 v \u22652 \u22122u \u00b7 v |u||v| . This is reduced to 2|u||v| \u22122u \u00b7 v \u22652 \u22122u \u00b7 v |u||v| . We denote |u||v| = x, u \u00b7 v = y, then the above can be represented as (x \u22121)(x \u2212y) \u22650. This inequality is true since |u|, |v| \u22651 and x \u2265y by the Cauchy-Schwarz inequality. Now we estimate \f \f \f ad |u| \u2212bd |v| \f \f \f. Since all hyperplanes intersect B(0, 1), one has |ad| |u| and |bd| |v| , i.e the distances from the origin to the hyperplanes \u03b31, \u03b32, are not larger than 1. Then \f \f \f \f ad |u| \u2212bd |v| \f \f \f \f = \f \f \f \f ad|v| \u2212ad|u| + ad|u| \u2212bd|u| |u||v| \f \f \f \f \u2264|u \u2212v| + |ad \u2212bd|. 5 \fThis completes the proof. Lemma 2.2. Let \u03a0 be a set of \u03b4-separated (\u03b4, t, C\u03c0)-hyperplanes intersecting B(0, 1). Then the set of dual points is c\u03b4-separated (\u03b4, t, c\u2032)-set for some absolute constants c, c\u2032 > 0. Proof. We denote \u03a0\u2217the set of dual points of hyperplanes \u03a0. It is immediate that the set \u03a0\u2217is c\u03b4-separated by Lemma 2.1. Hence, we only need to prove that, for arbitrary ball B(x, r), we have |\u03a0\u2217\u2229B(x, r)|\u03b4 \u2264c\u2032rt|\u03a0\u2217|\u03b4, \u03b4 \u2264r \u22641, for some constant c\u2032 that will be chosen later. It is su\ufb03cient to prove that | \b x\u2032 \u2208\u03a0\u2217: d(x\u2032, x) \u2264r \t | \u2264c\u2032rt|\u03a0\u2217|\u03b4, \u03b4 \u2264r \u22641. As mentioned earlier, the map D is bilipschitz onto its image, we have | {x\u2032 \u2208\u03a0\u2217: d(x\u2032, x) \u2264r} | is at most \f \f\b D(x\u2032) \u2208\u03a0 : dA(D(x\u2032), D(x)) \u2264KDr \t\f \f \u2264C\u03c0Kt Drt|\u03a0\u2217|\u03b4, \u03b4 \u2264r \u22641. Choose c\u2032 = C\u03c0Kt D, then the lemma follows. Note that Kt D can be replaced by some constant that does not depend on t since t \u2208(0, d). 2.1 Sobolev bounds for generalized Radon transforms and consequences In this section, we recall some known results that make use the boundedness of general Radon transforms. Let g: Rd \u2192R be a Schwartz function, t \u2208R, \u03c8: Rd \u00d7 Rd \u2192R be a smooth cut-o\ufb00 function, and \u03a8(x, y) : Rd \u00d7 Rd \u2192R be a smooth function with some suitable assumptions. We de\ufb01ne T\u03a8tg(x) := Z {\u03a8(x,y)=t} g(y)\u03c8(x, y)d\u03c3x,t(y), where d\u03c3x,t is the Lebesgue measure on the set {y: \u03a8(x, y) = t}. We denote the usual L2-Sobolev space of L2 functions with s generalized derivatives in L2(Rd) by L2 s(Rd). The following theorem was proved by Eswarathasan, Iosevich and Taylor in [3]. Theorem 2.3 ([3], Proposition 2.2). Let E \u2282Rd be a compact set with dimH(E) = \u03b1, and \u00b5 be the corresponding Frostman measure on E. Assume that T\u03a8t maps L2 to L2 s with constants uniform in a small neighborhood of t, and d \u2212s < \u03b1 < d. Then we have \u00b5 \u00d7 \u00b5{(x, y) \u2208E \u00d7 E : t \u2264|\u03a8(x, y)| \u2264t + \u01eb} \u2272\u01eb. Remark 2.1. It can be checked from Eswarathasan-Iosevich-Taylor\u2019s proof that the same result holds for two di\ufb00erent sets E \u00d7 F, namely, \u00b5E \u00d7 \u00b5F{(x, y) \u2208E \u00d7 F : t \u2264|\u03a8(x, y)| \u2264t + \u01eb} \u2272\u01eb, if T\u03a8t maps L2 to L2 s with d \u2212s < \u03b1, \u03b2 < d, where \u03b1 = dimH(E) and \u03b2 = dimH(F). Notice also that the sets E and F are not required to be A-D regular. Moreover in the proofs, the only thing that is needed associated with the set E is the existence of a probably measure \u00b5 supported on E that satis\ufb01es \u00b5(B(x, r)) \u2272r\u03b1, the same applies for F. To apply the above theorem in the proof of Theorem 1.4, we need to recall a celebrated result of Phong and Stein [12] stating that the operator T\u03a8t is uniformly bounded from L2 to L2 s on a small 6 \fneighborhood of t with s = d\u22121 2 if the so-called Phong-Stein rotational curvature condition det   0 \u2207x\u03a8 \u2212\u2207y\u03a8 \u22022\u03a8 \u2202xi\u2202yj ! \u0338= 0 (4) holds on the set {(x, y): \u03a8(x, y) = t}. This and Remark 2.1 imply the following theorem. Theorem 2.4. Let \u00b5E, \u00b5F be probability measures on compact sets E, F \u2282Rd, respectively, satisfying \u00b5E(B(x, r)) \u2272r\u03b1, \u00b5F(B(x, r)) \u2272r\u03b2, for all x \u2208Rd and r > 0. Suppose that the Phong-Stein rotational curvature condition (4) holds for the function \u03a8, and d \u2212(d \u22121)/2 < \u03b1, \u03b2 < d. Then, for \u01eb > 0, we have \u00b5E \u00d7 \u00b5F{(x, y) \u2208E \u00d7 F : |\u03a8(x, y)| \u2264\u01eb} \u2272\u01eb. 2.2 Proof of Theorem 1.4 Let \u03a0\u2217be the set of dual points corresponding to planes in \u03a0. As we proved in Lemma 2.2 that this set is c\u03b4-separated (\u03b4, t, c\u2032)-set. We \ufb01rst de\ufb01ne two probability measures on P(\u03b4) and \u03a0\u2217(c\u03b4), denoted by \u00b5P and \u00b5\u03a0, respectively, as follows: \u00b5P (X) = |X \u2229P(\u03b4)| |P(\u03b4)| , and \u00b5\u03a0\u2217(X) = |X \u2229\u03a0\u2217(c\u03b4)| |\u03a0\u2217(c\u03b4)| . Since P is (\u03b4, s, CP )-set and \u03a0\u2217is (c\u03b4, t, c\u2032)-set, we have \u00b5P(B(x, r)) \u2272rs, and \u00b5\u03a0\u2217(B(x, r)) \u2272rt, for all x \u2208Rd and r \u2265\u03b4. For r \u2264\u03b4, we have |B(x, r) \u2229P(\u03b4)| \u2272rd. On the other hand, we have |P(\u03b4)| \u2273\u03b4d\u2212s, which follows from the assumption that P is a (\u03b4, s, CP )-set. For r \u2264\u03b4 and s < d, we have rd\u2212s \u2264\u03b4d\u2212s, this means that \u00b5P (B(x, r)) \u2272 rd \u03b4d\u2212s \u2272rs. The same holds for \u03a0\u2217. In other words, the two probability measures \u00b5P and \u00b5\u03a0\u2217are Frostman measures with exponents s and t, respectively. To proceed further, we may assume that all hyperplanes are de\ufb01ned by the equation of the form a1x1 + \u00b7 \u00b7 \u00b7 + ad\u22121xd\u22121 + ad = xd. (5) This gives that the dual points are of the form (a1, . . . , ad\u22121, 1). De\ufb01ne \u03a8(x1, . . . , xd, a1, . . . , ad) = a1x1 + \u00b7 \u00b7 \u00b7 + ad\u22121xd\u22121 \u2212xd + ad. 7 \fA direct computation shows that this function satis\ufb01es the curvature condition (4). We now observe that (x1, . . . , xd) \u2208\u03c0(C\u03b4), where \u03c0 is de\ufb01ned by (5), if |\u03a8(x1, . . . , xd, a1, . . . , ad)| \u2264C\u03b4. On the other hand, if |\u03a8(x1, . . . , xd, a1, . . . , ad)| \u2264C\u03b4, then |\u03a8(U, V )| \u2272\u03b4 for all U \u2208B((x1, . . . , xd), \u03b4) and V \u2208B((a1, . . . , ad), c\u03b4), when \u03b4 is small enough. This infers that 1 |P||\u03a0||I\u03b4(P, \u03a0)| \u2264\u00b5P \u00d7 \u00b5\u03a0\u2217{(U, V ) \u2208P(\u03b4) \u00d7 \u03a0\u2217(c\u03b4): |\u03a8(U, V )| \u2272\u03b4} . Therefore, our result is reduced to show the following. \u00b5P \u00d7 \u00b5\u03a0\u2217{(U, V ) \u2208P(\u03b4) \u00d7 \u03a0\u2217(c\u03b4): |\u03a8(U, V )| \u2272\u03b4} \u2272\u03b4, which follows from Theorem 2.4. 3 Proof of Theorem 1.5 In this section, we present an elementary argument to study the incidence problem. For p \u2208P, let I(p) be the set of hyperplanes \u03c0 \u2208\u03a0 such that p \u2208\u03c0(C\u03b4). We observe that IC\u03b4(P, \u03a0) = X p\u2208P |I(p)|. Thus, for x, y > 0 and x \u2265y, by the H\u00a8 older inequality, we have IC\u03b4(P, \u03a0) \u2264|P| x x+y  X p |I(p)| x+y y ! y x+y . This implies that IC\u03b4(P, \u03a0)x+y \u2264|P|x  X p |I(p)| x+y y !y . To proceed further, we need to estimate the sum P p |I(p)|1+x/y. We have X p\u2208P |I(p)|1+x/y = X p\u2208P X \u03c0 : p\u2208\u03c0(C\u03b4) |I(p)|x/y = X \u03c0\u2208\u03a0 J(\u03c0), here J(\u03c0) = P p\u2208\u03c0(C\u03b4) |I(p)|x/y, which can be represented as follows J(\u03c0) = X p\u2208\u03c0(C\u03b4)  X i |I(p) \u2229J2i\u03b4(\u03c0)| !x/y \u2272 X p\u2208\u03c0(C\u03b4) log \u03b4\u22121 X i=1 |I(p) \u2229J2i\u03b4(\u03c0)|x/y, where J2i\u03b4(\u03c0) is the set of hyperplanes \u03c0\u2032 such that dA(\u03c0, \u03c0\u2032) \u223c2i\u03b4. Since the set of planes is (\u03b4, t, C\u03a0), we have |J2i\u03b4(\u03c0)| \u2272(2i\u03b4)t|\u03a0|. Lemma 3.1. Let \u03c0 and \u03c0\u2032 be two hyperplanes in A(d, d \u22121). Assume these two planes both intersect the unit ball B(0, 1) and dA(\u03c0, \u03c0\u2032) = w > \u03b4, then the intersection \u03c0(\u03b4) \u2229\u03c0\u2032(\u03b4) \u2229B(0, 1) 8 \fcan be covered by at most \u03b4\u2212(d\u22122) cubes of parameters \u03b4 w \u00d7 \u03b4 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 \u03b4 in Rd. Proof. The proof is essentially a combination of a number of results from [6]. Let e0 = (0, \u00b7 \u00b7 \u00b7 , 0) and {e1, \u00b7 \u00b7 \u00b7 , ed} be the standard basis of Rd. If we denote H0 = \u27e8ed\u27e9then H0 is a line and \u03c0 \u2229H0 is a point of the form (0, \u00b7 \u00b7 \u00b7 , 0, a0). We also denote Hi = ei + H0 so that \u03c0 \u2229Hi is also a point of the form (0, \u00b7 \u00b7 \u00b7 , 1, \u00b7 \u00b7 \u00b7 , ai). Let bi = ai \u2212a0, then we refer to a0 as the vertical intercept and bi as the slopes of hyperplane \u03c0. For each hyperplane \u03c0, we associate it to a point x = x(\u03c0) = (a0, b1, \u00b7 \u00b7 \u00b7 , bd\u22121) \u2208Rd and note that the map \u03d5: \u03a0 7\u2192x(\u03a0) is well-de\ufb01ned and injective. The space generated by this map is called code space. We endow this space with the maximum metric, namely, ||x \u2212x\u2032|| := max \u0012 |a0 \u2212a\u2032 0|, max i=1,...,d\u22121 \u0000|bi \u2212b\u2032 i| \u0001\u0013 . As noted in [6, Remark 4.2], the maximum metric on the code space is strongly equivalent to the metric between hyperplanes dA, in the sense that ||\u03d5(\u03c0) \u2212\u03d5(\u03c0\u2032)|| \u223cdA(\u03c0, \u03c0\u2032). Fix two hyperplanes \u03c0, \u03c0\u2032, and denote their two corresponding codes by x(\u03c0) and x(\u03c0\u2032). Then we can present the plane \u03c0 by the equation a0 + b1t1 + \u00b7 \u00b7 \u00b7 + bd\u22121td\u22121 = td, (t1, . . . , td\u22121) \u2208Rd\u22121, where x(\u03c0) = (a0, b1, \u00b7 \u00b7 \u00b7 , bd\u22121). Similarly, we can de\ufb01ne \u03c0\u2032 from its code coordinate. As presented in the proof of [6, Lemma 4.3] that there is a constant c independent on \u03b4 such that: \u03c0(\u03b4) \u2282\u03c0 + ({0} \u00d7 (\u2212c\u03b4, c\u03b4)). Therefore, we have \u03c0(\u03b4) \u2229\u03c0\u2032(\u03b4) \u2229C \u2282 n (t, u) \u2208Rd : u \u2208B(td, c\u03b4) \u2229B(t\u2032 d, c\u03b4) o , where C = Cd\u22121 \u00d7 L1 and Cd\u22121 is the convex hull of (0, . . . , 0), e1, . . . , ed\u22121, and L1 is the unit segment centered at the origin in < ed >. If |a0 \u2212a\u2032 0| > maxi=1,...,d\u22121 (|bi \u2212b\u2032 i|) + D\u03b4, then choosing D = 2c implies B(td, c\u03b4) \u2229B(t\u2032 d, c\u03b4) is empty. This infers that \u03c0(\u03b4) \u2229\u03c0\u2032(\u03b4) \u2229C = \u2205. If maxi=1,...,d\u22121(|bi \u2212b\u2032 i|) > 0 and B(td, c\u03b4) \u2229B(t\u2032 d, c\u03b4) \u0338= \u2205, we set N = {t \u2208Cd\u22121 : |td \u2212t\u2032 d| < 2c\u03b4}, then we have N = {t \u2208Cd\u22121 : p\u2212(t) \u2264td\u22121 \u2264p+(t)} where p\u2212(t) = p\u2212(t1, \u00b7 \u00b7 \u00b7 , td\u22122) = \u22122c\u03b4 \u2212(a0 \u2212a\u2032 0) \u2212Pd\u22122 i=1 ti(bi \u2212b\u2032 i) bd\u22121 \u2212b\u2032 d\u22121 , and p+(t) = p+(t1, \u00b7 \u00b7 \u00b7 , td\u22122) = 2c\u03b4 \u2212(a0 \u2212a\u2032 0) \u2212Pd\u22122 i=1 ti(bi \u2212b\u2032 i) bd\u22121 \u2212b\u2032 d\u22121 . This implies that N is the intersection of Cd\u22121 and the strip between two parallel hyperplanes {td\u22121 = p\u2212(t)} and {td\u22121 = p+(t)}. By a direct computation, the distance between these two 9 \fhyperplanes is equal to d(p\u2212(t), p+(t)) = 2c\u03b4 qPd\u22122 i=1 (bi \u2212b\u2032 i)2 \u2272 \u03b4 maxi=1,...,d\u22121 |bi \u2212b\u2032 i|. Hence, N is contained in a rectangular box that has the shortest side of length d \u2272\u03b4/w and the other d \u22122 sides of length at most diam(C) = \u221a 2. To conclude the proof, we do a rotation if needed to assume that the hyperplane \u03c0 has ed as normal vector. This gives \u03c0 : f(t) = a0. The hyperplane \u03c0\u2032 has the equation of the form \u03c0\u2032 : g(t) = a\u2032 0 + b\u2032 1t1 + \u00b7 \u00b7 \u00b7 + b\u2032 d\u22121td\u22121. Under these hypotheses, if (t, u) and (t\u2032, u\u2032) are two elements inside \u03c0(\u03b4) \u2229\u03c0\u2032(\u03b4) \u2229C, then triangle inequality gives |u \u2212u\u2032| = |u \u2212f(t) + f(t\u2032) \u2212u\u2032| \u2264|u \u2212f(t)| + |u \u2212f(t\u2032)| < 2\u03b4 Therefore, the d-th coordinate of the intersection part is contained in a box of dimension \u2272\u03b4. In other words, the intersection \u03c0(\u03b4) \u2229\u03c0\u2032(\u03b4) \u2229C can be covered by roughly \u03b42\u2212d dyadic boxes of size \u03b4 w \u00d7 \u03b4 \u00d7 \u00b7 \u00b7 \u00b7 \u03b4 | {z } d \u22121 times . This completes the proof. Lemma 3.2. Fix \u03c0 \u2208\u03a0. For any \u03b4 < w \u226a1, we have X p\u2208\u03c0(c\u03b4) |I(p) \u2229Jw(\u03c0)| \u2272|Jw(\u03c0)| \u00b7 |P|\u03b4s \u00b7 1 wmin{s,1} \u00b7 1 \u03b4d\u22122 . Proof. Fix \u03c0 \u2208Jw(\u03c0), then it follows from Lemma 3.1 that \u03c0 \u2229\u03c0\u2032 is contained in the union of \u03b42\u2212d boxes with parameters \u03b4 w \u00d7 \u03b4 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 \u03b4. This means that the diameter of each box is \u223c\u03b4/w. We now bound the above sum in two ways: Since P is \u03b4-separated, each box contains at most \u22721 w elements from P. This gives X p\u2208\u03c0(c\u03b4) |I(p) \u2229Jw(\u03c0)| \u2272|Jw(\u03c0)| \u00b7 1 w \u00b7 1 \u03b4d\u22122 . We also observe that each box is contained in a ball of radius \u03b4/w, this infers each box contains at most CP |P|\u03b4s 1 ws balls of P since P is (\u03b4, s, CP ) set. In total, one has X p\u2208\u03c0(c\u03b4) |I(p) \u2229Jw(\u03c0)| \u2272|Jw(\u03c0)| \u00b7 |P|\u03b4s \u00b7 1 ws \u00b7 1 \u03b4d\u22122 . Note that by assumption we have |P|\u03b4s \u22651 so that combining these two estimates, we get X p\u2208\u03c0(c\u03b4) |I(p) \u2229Jw(\u03c0)| \u2272|Jw(\u03c0)| \u00b7 |P|\u03b4s \u00b7 1 wmin{s,1} \u00b7 1 \u03b4d\u22122 . 10 \fThis completes the proof of the lemma. We now continue the proof of the incidence estimate. In particular, J(\u03c0) \u2272 X p\u2208\u03c0(C\u03b4) X i |I(p) \u2229J2i\u03b4(\u03c0)|x/y \u2272 X i X p\u2208\u03c0(C\u03b4) |I(p) \u2229J2i\u03b4(\u03c0)| \u00b7 |J2i\u03b4(\u03c0)|x/y\u22121 \u2272 X i |(2i\u03b4)t|\u03a0||x/y \u00b7 |P|\u03b4s \u00b7 1 (2i\u03b4)min{s,1} \u00b7 1 \u03b4d\u22122 . We now fall into the following cases: If s \u22651 and tx/y \u22651, then J(\u03c0) \u2272|\u03a0|x/y \u00b7 |P|\u03b4s \u00b7 1 \u03b4d\u22122 \u00b7 X i (2i\u03b4) tx y \u22121 \u2272|\u03a0|x/y \u00b7 |P|\u03b4s \u00b7 1 \u03b4d\u22122 . If s \u22651 and tx/y < 1, then J(\u03c0) \u2272|\u03a0|x/y \u00b7 |P|\u03b4s \u00b7 1 \u03b4d\u22122 \u00b7 X i (2i\u03b4) tx y \u22121 \u2272|\u03a0|x/y \u00b7 |P|\u03b4s+ tx y \u22121 \u00b7 1 \u03b4d\u22122 . If s < 1 and tx/y \u2265s, then J(\u03c0) \u2272|\u03a0|x/y \u00b7 |P|\u03b4s \u00b7 1 \u03b4d\u22122 \u00b7 X i (2i\u03b4) tx y \u2212s \u2272|\u03a0|x/y \u00b7 |P|\u03b4s \u00b7 1 \u03b4d\u22122 . If s < 1 and tx/y < s, then J(\u03c0) \u2272|\u03a0|x/y \u00b7 |P|\u03b4s \u00b7 1 \u03b4d\u22122 \u00b7 X i (2i\u03b4) tx y \u2212s \u2272|\u03a0|x/y \u00b7 |P|\u03b4 tx y \u2212d+2. Plugging these estimates into IC\u03b4(P, \u03a0), one has Case 1: If s \u22651 and tx/y \u22651, then IC\u03b4(P, \u03a0) \u2272|P||\u03a0|\u03b4 y x+y (s\u2212d+2). If s \u2212d + 2 < 0 then we only have the trivial upper bound. Thus, this upper bound is valid in the range s \u2212d + 2 > 0. By choosing x = y, we obtain the upper bound IC\u03b4(P, \u03a0) \u2272|P||\u03a0|\u03b4 1 2(s\u2212d+2). Case 2: If s \u22651 and tx/y < 1, then IC\u03b4(P, \u03a0) \u2272|P||\u03a0|\u03b4 y x+y \u0010 s+ tx y \u2212d+1 \u0011 . We observe that t \u22641 since x \u2265y. If s \u2212d + 2 < 0, then we only have the trivial upper bound. If 11 \fs \u2212d + 2 > 0, then we can choose y = xt + \u01eb, \u01eb > 0, to get the upper bound IC\u03b4(P, \u03a0) \u2272|P||\u03a0|\u03b4 t 1+t (s\u2212d+2)\u2212\u01eb. Case 3: If s < 1 and tx/y \u2265s, then IC\u03b4(P, \u03a0) \u2272|P||\u03a0|\u03b4 y x+y (s\u2212d+2). In this case, since s < 1 and d \u22653, we have s \u2212d + 2 is always negative, so only trivial upper bound is obtained. Case 4: If s < 1 and tx/y < s, then IC\u03b4(P, \u03a0) \u2272|P||\u03a0|\u03b4 y x+y \u0010 tx y \u2212d+2 \u0011 . Since tx/y < s, s < 1, and d \u22653, we only have the trivial upper bound in this case. Putting these cases together, Theorem 1.5 follows. 4 Sharpness of Theorem 1.4 In this section we provide an example to show the sharpness of our result in Theorem 1.4. The construction is mainly adapted from the two dimensional example due to Fu and Ren in [4]. For the reader\u2019s convenience, we sketch the ideas here. We \ufb01rst recall that Construction 4 in [4] shows that in the plane when s + t \u22653, the incidence between balls and tubes is \u223c\u03b4\u2212(s+t\u22121) = \u03b4|P||T |. We will see that their example can be extended to higher dimensions which gives us a sharp upper bound. Note that the incidences results of Fu and Ren [4] in the plane are applied to the sets of balls and tubes satisfying De\ufb01nition 1.2 (Katz-Tao (\u03b4, s) sets). However, as we mentioned before, when the set is of size \u223c\u03b4\u2212s then De\ufb01nitions 1.1 and 1.2 are equivalent. Moreover, in their sharpness example, what they constructed are actually (\u03b4, s, C) sets. Therefore, we can extend their construction to higher dimensional spaces. In the rest, we denote D = \u03b4\u22121 for convenience. 4.1 Construction in R3 The idea is to construct a con\ufb01guration such that the intersection between this con\ufb01guration and the plane Oxy is exactly Construction 4 in [4]. Thus, we can reduce to the two-dimensional case when we consider the intersection between \u03b4-hyperplanes and \u03b4-balls with Oxy. Then we move the plane Oxy vertically by spacing 2\u03b4 (two consecutive planes have distance 2\u03b4) to get the desired bound. We consider balls of form: (x \u2212a1)2 + (y \u2212a2)2 + z2 = \u03b42. Clearly these are \u03b4-balls in R3 whose intersection with Oxy are also \u03b4-balls in dimension two. For each tube in Construction 4 in [4], we just extend the tube vertically to the R3 which becomes a plane. Thus, if we put balls by spacing 2\u03b4 in R3 (the centers of two consecutive balls have distance of 2\u03b4), then the incidence between these balls and planes is \u223c\u03b4\u2212(s+t\u22121) \u00b7\u03b4\u22121 = \u03b4 \u00b7\u03b4\u2212s\u22121 \u00b7\u03b4\u2212t which is equal to \u03b4|P||\u03a0|. 12 \fThe next step is to check that the set of balls and hyperplanes constructed above satisfy De\ufb01nition 1.1. Since the hyperplanes are just the tubes expanding vertically, the number of \u03b4-hyperplanes is equal to that of \u03b4-tubes, i.e, \u223cDt. Moreover, as already shown in the paper of Fu and Ren [4] (page 6) that the set of \u03b4-tubes is a (\u03b4, t, C) set which gives that our set of \u03b4-hyperplanes is also a (\u03b4, t, C) set. Now for each copy of Oxy, there are \u223cDs balls, as stated in [4], and there are \u03b4\u22121 copies in total. So |P| \u223cDs+1. Moreover, as shown in [4], for each copy of Oxy, a ball of radius w in the plane contains at most Cws\u03b4\u2212s \u03b4-balls. So, a ball Bw in R3 contains at most Cws\u03b4\u2212s\u22121 \u03b4-balls in P. This gives that |{p \u2208P : p \u2282Bw}| \u2264Cws\u03b4\u2212s\u22121 \u223cCws|P|. Therefore, the set P is a (\u03b4, s, C)-set. 4.2 Construction in Rd, d \u22654 For dimension d > 3, the construction is inductively constructed. For instance, we consider d = 4. We directly extend each hyperplane from above 3-dimensional construction to a hyperplane in R4 by just adding a variable x4. Precisely, assume a hyperplane in above 3-dimensional construction is de\ufb01ned by {(x1, x2, x3) : ax1 + bx2 + cx3 = e}, then the extension in R4 is de\ufb01ned by {(x1, x2, x3, x4) : ax1 + bx2 + cx3 + 0x4 = e}. We now consider balls de\ufb01ned by the equation: (x1 \u2212a1)2 + (x2 \u2212a2)2 + x2 3 + x2 4 = \u03b42. These balls are moved by spacing 2\u03b4 in x3 and x4 directions. Thus, we see that when we \ufb01x the last coordinate x4, the number of incidences between these hyperplanes and balls is the same as we constructed above in 3-dimension which is \u03b4\u03b4\u2212s\u22121\u03b4\u2212t. Moreover, we have \u03b4\u22121 number of copies in x4 direction which implies that the number of incidences is \u03b4\u03b4\u2212s\u22121\u03b4\u2212t\u03b4\u22121 = \u03b4|P||\u03a0|. It is not di\ufb03cult to prove that the sets of balls and hyperplanes are (\u03b4, s, C) and (\u03b4, t, C) sets, respectively. For other dimensions d > 4, the construction works in the same way. Thus, the number of incidences I(P, \u03a0) \u223c|{incidences in the plane}| \u00b7 \u03b42\u2212d \u223c\u03b4|P||\u03a0|. For the sharpness of Theorem 1.5, one might think of the same idea by extending Constructions 1, 2, 3 in [4] to higher dimensions, but it does not match the upper bounds in Theorem 1.5 at least in the way we tried. 5 Acknowledgements T. Pham would like to thank the Vietnam Institute for Advanced Study in Mathematics (VIASM) for the hospitality and for the excellent working condition. C.-Y. Shen was partially supported by NSTC grant 111-2115-M-002-010-MY5."
+                },
+                {
+                    "url": "http://arxiv.org/abs/2303.07878v1",
+                    "title": "VC-dimension and pseudo-random graphs",
+                    "abstract": "Let $G$ be a graph and $U\\subset V(G)$ be a set of vertices. For each $v\\in\nU$, let $h_v\\colon U\\to \\{0, 1\\}$ be the function defined by\n\\[h_v(u)=\\begin{cases} &1 ~\\mbox{if}~u\\sim v, u\\in U\\\\&0 ~\\mbox{if}~u\\not\\sim\nv, u\\in U\\end{cases},\\] and set $\\mathcal{H}(U):=\\{h_v\\colon v\\in U\\}$. The\nfirst purpose of this paper is to study the following question: What families\nof graphs $G$ and what conditions on $U$ do we need so that the VC-dimension of\n$\\mathcal{H}(U)$ can be determined? We show that if $G$ is a pseudo-random\ngraph, then under some mild conditions, the VC dimension of $\\mathcal{H}(U)$\ncan be bounded from below. Specific cases of this theorem recover and improve\nprevious results on VC-dimension of functions defined by the well-studied\ndistance and dot-product graphs over a finite field.",
+                    "authors": "Thang Pham, Steven Senger, Michael Tait, Nguyen Thu-Huyen",
+                    "published": "2023-03-14",
+                    "updated": "2023-03-14",
+                    "primary_cat": "math.CO",
+                    "cats": [
+                        "math.CO"
+                    ],
+                    "main_content": "Introduction In this paper, we study the VC-dimension of sets of functions that are de\ufb01ned by graph adjacency. We start with the requisite de\ufb01nitions. De\ufb01nition 1.1. Let H be a collection of functions from V to {0, 1}. We say that H shatters a \ufb01nite set X \u2282V if the restriction of H to X yields every possible function from X to {0, 1}. De\ufb01nition 1.2. Let H be a collection of functions from V to {0, 1}. The Vapnik-Chervonenkis dimension (in short, VC-dimension) of H is d if any only if there exists a set X \u2282V of size d that is shattered by H, and no subset of X of size d + 1 is shattered by H. It is mentioned in the book [9] that \u201cthe VC-dimension can be determined without great di\ufb03culty in several simple cases, such as for half-spaces or balls in Rd, but for only slightly more complicated families its computation becomes challenging.\u201d Before stating our main results, we need to set up notation as follows. Let G = (V, E) be a graph with the vertex set V and the edge set E. For two vertices u, v \u2208V , by u \u223cv, we mean there is an edge between u and v. Let U \u2282V be a set of vertices. For each v \u2208U, let hv : U \u2192{0, 1} be the function de\ufb01ned by hv(u) = ( 1 if u \u223cv, u \u2208U 0 if u \u0338\u223cv, u \u2208U , \u2217University of Science, Vietnam National University, Hanoi. Email: phamanhthang.vnu@gmail.com \u2020Department of Mathematics, Missouri State University. Email: StevenSenger@MissouriState.edu \u2021Department of Mathematics and Statistics, Villanova University. Email: michael.tait@villanova.edu \u00a7Fulbright University Vietnam. Email: huyen.nguyen.190033@student.fulbright.edu.vn 1 arXiv:2303.07878v1  [math.CO]  14 Mar 2023 \fand set H(U) := {hv : v \u2208U}. The following question appears to us to be natural and fundamental. Question 1.3. For which families of graphs and under what conditions on U can we determine the VC-dimension of H(U)? One of the primary motivations for this question comes from a series of recent papers [2, 4, 6, 10] in which the authors studied graphs G with V (G) = Ft q, where Fq is the \ufb01nite \ufb01eld of order q, and there is an edge between two vertices x and y if P(x, y) = 1 for some polynomials P. More precisely, if t = 2 and P(x, y) = (x1 \u2212y1)2 + (x2 \u2212y2)2, then Fitzpatrick, Iosevich, McDonald, and Wyman [4] showed that for any subset U \u2282F2 q of size at least Cq15/8, the set H(U) has VC-dimension equal to 3. They also showed that for |U| \u2265Cq3/2 the set H(U) has VC-dimension at least 2. They asked speci\ufb01cally about the gap between these two exponents and if one can determine the VC-dimension for sets of size o(q15/8). When t = 3 and P(x, y) = x1y1 + x2y2 + x3y3, Iosevich, McDonald, and Sun [6] showed that in this graph any subset U \u2282F3 q of size at least Cq11/4 has the property that H(U) has VC-dimension equal to 3, and that any set with |U| \u2265Cq5/2 has VC-dimension at least 2. They similarly remarked that they \u201cdo not know to what extent the exponent 11 4 ... and ... 5 2 are sharp, but we know that neither exponent can fall below 2.\u201d Our main goal in this paper is to develop a framework for these questions. Ideally, we would like to be able to prove theorems like the concrete examples given above as speci\ufb01c applications of a general theorem. We make partial progress towards this goal. More precisely, we will address Question 1.3 for (n, d, \u03bb)-graphs, i.e. graphs with n vertices, regular of degree d, and the second largest eigenvalue is at most \u03bb in absolute value. Our main results can be stated as follows. Theorem 1.4. Let G be an (n, d, \u03bb)-graph with d = o(n). Let U be a vertex set. Assume that \u03bbn/d = o(|U|), then the VC-dimension of H(U) is at least 2. If more conditions are allowed, then one can be guaranteed a larger dimension in the next theorem. Although in the next theorem the dimension is increasing by only one from Theorem 1.4, the proof becomes much more di\ufb03cult. Theorem 1.5. Let G be an (n, d, \u03bb)-graph with d = o(n). Let U be a vertex set. Assume that the following properties hold. 1. Given any three vertices v1, v2, v3 \u2208U, we can \ufb01nd three vertices u1, u2, u3 \u2208U such that ui \u223cvj if and only if i = j, 2. The size of U satis\ufb01es |U| \u2265C max n \u03bb2/3(n/d)2, \u03bb(n/d)13/7o . where C is an absolute constant. Then the VC-dimension of H(U) is at least 3. We make some remarks before proceeding. First, since |H(U)| = |U|, the trivial upper bound gives that the VC-dimension is at most log2(|U|), and sets with VC-dimension 2 or 3 are far from this. Second, when showing that the VC-dimension of a set is at least 3, we require condition 1 in the hypotheses of Theorem 1.5. It would be very interesting to get rid of this condition or replace it with a more natural one that applies to (n, d, \u03bb)-graphs in a more general way. Finally, in all of the previously studied graphs de\ufb01ned by distances or dot-products, there are geometric reasons that make \ufb01nding a corresponding upper bound on the VC-dimension easy. We do not know how 2 \fto give upper bounds on VC-dimension in a general way for (n, d, \u03bb)-graphs. The main idea to prove Theorems 1.4 and 1.5 is to \ufb01nd a subset of 2 or 3 vertices respectively that are shattered by a large enough subset U. From the way the functions are de\ufb01ned, we must \ufb01nd sets of vertices with prescribed adjacencies and non-adjacencies; the backbone of our proof will be to show that in a suitably large subset of a pseudo-random graph, we can \ufb01nd and count various subgraphs. We \ufb01nd these results interesting in their own right, as \ufb01nding or counting certain subgraphs in pseudo-random graphs has a long history of study. For example, a representative result is the following classical theorem of Alon [7, Theorem 4.10]. Theorem 1.6. Let H be a \ufb01xed graph with r edges, s vertices and maximum degree \u2206, and let G be an (n, d, \u03bb)-graph with d \u22640.9n. Then for any subset U with \u03bb \u0000 n d \u0001\u2206= o(|U|), the number of copies of H in U is (1 + o(1)) |U|s |Aut(H)| \u0012d n \u0013r We will state our subgraph counting theorems in the coming sections and give some discussion of them, including comparison with Theorem 1.6, in Section 3. Instead of counting subgraphs exactly, it will be more useful for us to instead count homomorphisms from a graph H to the graph induced by a subset of vertices U in our graph. Given a subset of vertices U \u2282V (G) and a \ufb01xed subgraph H, we will use the notation H(U) to denote the number of homomorphisms from H to the induced subgraph G[U]. For example, if H is a cycle on k vertices, then Ck(U) denotes the number of sequences of vertices {v1, \u00b7 \u00b7 \u00b7 , vk} such that vi \u2208U and vi \u223cvi+1 for 1 \u2264i \u2264k \u22121 and vi \u223cvk. Notice that this is counting labeled and possibly degenerate cycles. 1.1 Applications We now present some applications of Theorems 1.4 and 1.5. There are many (n, d, \u03bb)-graphs that can be computed explicitly in the literature, such as Cayley graphs. In this section, we only emphasize some particular ones, viz., dot-product graphs and distance graphs. Let G = (V, E) be the dot-product graph in Ft q de\ufb01ned by V = Ft q \\ (0, . . . , 0), and there is an edge between two vertices (x1, x2, . . . , xt) and (y1, y2, . . . , yt) if x1y1 + \u00b7 \u00b7 \u00b7 + xtyt = 1. It is well-known in the literature that this graph is an (n, d, \u03bb)-graph with n = qt \u22121, d \u223cqt\u22121 and \u03bb \u22642q t\u22121 2 , see [12, Theorem 8.1] for example. The following is the direct application of Theorems 1.4 and 1.5. Theorem 1.7. Let U \u2282Ft q be a subset of vertices in the dot-product graph. There is a constant C such that 1. If |U| \u2265Cq t+1 2 , then the dimension of H(U) is at least 2. 2. If |U| \u2265C max{q 7t+19 14 , qt\u22121}, then the dimension of H(U) is at least 3. In three dimensions, if the geometric properties are taken into consideration, then we have a better result, which improves earlier results in [6]. Theorem 1.8. Let U \u2282F3 q be a subset of vertices in the dot-product graph. There is a constant C such that 1. If |U| \u2265Cq2, then the dimension of H(U) is at least 2. 2. If |U| \u2265Cq5/2, then the dimension of H(U) is equal to 3. 3 \fThis theorem recovers the exponent 3/2 in dimension 2 from the paper [6] and improves the exponent 11/4 in dimension 3 from [6] down to 5/2. We now move to the graph de\ufb01ned by the distance function. Let G = (V, E) be the distance graph in Ft q de\ufb01ned by V = Ft q, and there is an edge between two vertices (x1, x2, . . . , xt) and (y1, y2, . . . , yt) if (x1 \u2212y1)2 + \u00b7 \u00b7 \u00b7 + (xt \u2212yt)2 = 1. It is well-known in the literature that this graph is an (n, d, \u03bb)-graph with n = qt, d \u223cqt\u22121, \u03bb \u22642q t\u22121 2 -graph, see [3, Sections 2\u20136] and [8, Section 3] for example. As above, we also have the following application in this setting. Theorem 1.9. Let U \u2282Ft q be a subset of vertices in the distance graph. There is a constant C such that 1. If |U| \u2265Cq t+1 2 , then the dimension of H(U) is at least 2. 2. If |U| \u2265C max{q 7t+19 14 , qt\u22121}, then the dimension of H(U) is at least 3. Compared to the result in [4] with t = 2, we recover their result to \ufb01nd VC-dimension 2 but we can see that we get a trivial result for VC-dimension 3 whereas in [4] the exponent 15/8 was given. This comes from the fact that in some speci\ufb01c settings, counting subgraph con\ufb01gurations in the next sections is unnecessary, and in the plane F2 q, the distance graph possesses some nice geometric properties, which implies a simpler proof with better exponents. Unlike the dot-product graph, we do not know how to improve the exponent 15/8, which is left as an open question. 1.2 Open questions We do not believe that Theorem 1.5 is sharp in general. In particular, one of our estimates on subcon\ufb01gurations (Theorem 3.2 below) involves a recursive step using Cauchy-Schwarz that could probably be improved in many cases, but this was the best estimate we found. In Section 3 we point out other con\ufb01gurations for which we believe it would be interesting to tighten our estimates. Our theorems give conditions to guarantee that the VC-dimension of an (n, d, \u03bb)-graph is at least 2 or 3. In [1], for each \ufb01xed D the authors give the threshold function for a random graph to have VC-dimension at least D. It would be interesting to do this in the setting of (n, d, \u03bb)-graphs: that is, to give conditions on d and \u03bb which would guarantee that the graph has VC-dimension at least D. New ideas would be needed to do this. Already to show a lower bound of VC-dimension 4 using the techniques in this paper is outside of our capabilities. Finally, in Theorem 1.8, we do not have a construction showing that the exponent 5/2 is best possible, but we believe that it could be, and it would be interesting to determine if this is correct. 1.3 Structure The paper is organized as follows. In Section 2 we collect preliminary lemmas that we will require during the proofs. In Section 3 we show that we can count or bound H(U) for various graphs H as long as U is large enough. In Sections 4 and 5, we show how to use these counting results to prove Theorems 1.4 and 1.5. Finally, in Section 6, we show our applications to the distance and dot-product graphs. Given two functions f, g : N \u2192N we will use the notation f \u226ag to mean that f = O(g) and f \u226bg to mean that f = \u2126(g). 4 \f2 Tools In this section we list the tools that we will need in our proofs. We will extensively use the following weighted version of the expander mixing lemma. As a historical note, the (unweighted) expander mixing lemma was proved at least as early as 1980 by Haemers in his PhD thesis ([5] Theorem 3.1.1) in the language of design theory. Lemma 2.1. Let G = (V, E) be an (n, d, \u03bb)-graph, and A be its adjacency matrix. For real f, g \u2208L2(V ), we have |\u27e8f, Ag\u27e9\u2212dnE(f)E(g)| \u2264\u03bb\u2225f\u22252\u2225g\u22252, where E(f) := 1 n X v\u2208V f(v), and ||f||2 2 = X v\u2208V |f(v)|2. At times, the classical expander mixing lemma will not be precise enough for our purposes. We will use the following lemma, \ufb01rst proved in [11], that is speci\ufb01c to tensor powers of graphs. Notice that the function f below has a di\ufb00erent domain than f above. Proposition 2.2. Let G be an (n, d, \u03bb)-graph. For two non-negative functions f, g: V \u00d7 V \u2192R, we de\ufb01ne F(x) = P y f(x, y), G(z) = P w g(z, w), F \u2032(y) = P x f(x, y), and G\u2032(w) = P z g(z, w). Then we have \f \f \f \f \f X x\u223cz,y\u223cw f(x, y)g(z, w) \u2212d2 n2 ||f||1||g||1 \f \f \f \f \f \u2264\u03bb2||f||2||g||2 + d\u03bb n \u0000||F||2||G||2 + ||F \u2032||2||G\u2032||2 \u0001 . We will also need an extension of the expander mixing lemma to counting paths and cycles. The following can be proved using Lemma 2.1 or Proposition 2.2 and induction, see [11]. Proposition 2.3 (Proposition 3.5 in [11]). Let G be an (n, d, \u03bb)-graph, k \u22651 an integer, and U be a vertex set with \u03bb \u00b7 n d = o(|U|). Let Pk(U) denote the number of (labeled, possibly degenerate) paths with k edges in U. Then we have Pk(U) = (1 + o(1)) |U|k+1dk nk . Theorem 2.4 (Theorem 1.5 in [11]). Let G be an (n, d, \u03bb)-graph and U be a vertex set with \u03bb \u00b7 n d = o(|U|). Let Cm(U) denote the number of (labeled, possibly degenerate) cycles of length m with vertices in U. Then we have \f \f \f \fCm(U) \u2212|U|mdm nm \f \f \f \f = O \u0012\u03bb|U|m\u22121dm\u22121 nm\u22121 + \u03bbm\u22122|U|2d n \u0013 , The following is a straightforward technical lemma used in the arguments below, showing that subsets of pseudo-random graphs cannot have too many vertices of too large or small degree. Lemma 2.5. Let G be an (n, d, \u03bb)-graph. Assume U \u2282V (G) with \u03bb(n/d) = o(|U|), then there exists a subset U \u2032 \u2282U such that |U \u2032| = (1 \u2212o(1))|U| and |U|d 2n \u2264 X v\u2208U X u\u223cv 1 \u22642|U|d n , 5 \ffor any u \u2208U \u2032. Proof. Let L \u2282U be the subset of vertices of U with more than 2|U|d n neighbors in U and S the subset of vertices with fewer than 1 2 |U|d n neighbors in U. Then by Lemma 2.1 we have that |L| \u00b7 2|U|d n \u2264e(L, U) \u2264|L||U| d n + \u03bb p |L||U|, which implies that |L| \u2264\u03bb2n2 d2|U| = o(|U|), by the assumption on the size of U. Similarly, |S| = o(|U|) and taking U \u2032 = U \\ (S \u222aL) gives the result. 3 Counting con\ufb01gurations In this section, we count several con\ufb01gurations which we will need in order to prove Theorems 1.4 and 1.5. The most important con\ufb01gurations are pictured in Figure 1. Let H1 be the number of 5-tuples (x, y, z, u, v) \u2208U 5 such that x \u223cy, y \u223cz, z \u223cu, u \u223cv, u \u223cx. Let H2 be the number of 5-tuples (x, y, z, u, v) \u2208U 5 such that x \u223cu, u \u223cz, z \u223cy, x \u223cy, u \u223cv, v \u223cy That is, H2 is a copy of K2,3. Let H3 be the number of 7-tuples (x, y, z, u, v, u\u2032, x\u2032) \u2208U 7 such that x \u223cy, y \u223cz, z \u223cu, u \u223cx, u \u223cv, v \u223cu\u2032, u\u2032 \u223cz, u\u2032 \u223cx\u2032, x\u2032 \u223cy. We de\ufb01ne two more con\ufb01gurations which are related to H3. Let H+ 3 be the same as con\ufb01guration H3 with the additional condition that x \u223cu\u2032. And let H\u2212 3 be the same con\ufb01guration as H3 with the additional restriction that x and u\u2032 are the same vertex. That is, H\u2212 3 is 6 vertices x, y, z, u, v, x\u2032 in U satisfying x \u223cy, y \u223cz, z \u223cu, u \u223cx, x \u223cx\u2032, x\u2032 \u223cy, x \u223cv, v \u223cu, x \u223cz. In this section, we show that for large enough subsets U, there will be a subset U \u2032 \u2282U with |U \u2032| = (1 \u2212o(1))|U| such that we can estimate the number of con\ufb01gurations in U \u2032 precisely. We do not know if passing to the subset U \u2032 is necessary. However, we do this so that we can bound the number of K1,3 and K1,4 in our subsets. We will be interested in subsets U which satisfy \u03bb n d = o(|U|). In some regimes, sets of this size do not have the expected number of stars on 3 and 4 edges. For example, given an (n, d, \u03bb) graph with d = \u221an, if a set U has size n3/4, then we expect K1,4(U) to be \u0398 \u0000n7/4\u0001 . On the other hand, taking a set of n1/4 vertices and their neighborhoods gives a set U with at least \u2126 \u0000n9/4\u0001 copies of K1,4. Note that if we wanted to apply Theorem 1.6, 6 \fFigure 1: Con\ufb01gurations H1, H2, and H3 Figure 2: Con\ufb01gurations H+ 3 , H\u2212 3 , and H4 we would require \u03bb \u0000 n d \u00014 = o(|U|) which is much larger than the size of sets with which we wish to work. For our application to VC-dimension, passing to a subset U \u2032 does not weaken the result, and so for the trade-o\ufb00of weakening the theorems in this section, we obtain in some cases the best possible results for VC-dimension. Theorem 3.1. Let G be an (n, d, \u03bb)-graph and U a subset of vertices with \u03bbn/d = o(|U|). Then there is a subset U \u2032 \u2282U with |U \u2032| = (1 \u2212o(1))|U| such that \f \f \f \fH1(U \u2032) \u2212|U \u2032|5d5 n5 \f \f \f \f \u226a\u03bb2 |U \u2032|3d2 n2 + \u03bb|U \u2032|4d4 n4 . Proof. By Lemma 2.5, we may choose a subset U \u2032 \u2282U with |U \u2032| = (1 \u2212o(1))|U| such that no vertex in U \u2032 has degree more than 2|U|d n in U. De\ufb01ne f(x, y) to be the number of paths of length 2 of the form (x, y, u) \u2208(U \u2032)3 with x \u223cy, y \u223cu. De\ufb01ne g(z, w) = 1z\u223cw, where the notation 1z\u223cw denotes the function that returns a 1 if z \u223cw and 0 otherwise. Since ||f||1 is equal to the number of paths of length 2 with vertices in U \u2032, it follows from Proposition 2.3 that ||f||1 = P2(U \u2032) = (1 + o(1))|U \u2032|3d2 n2 and ||g||1 = P1(U \u2032) = (1 + o(1))|U \u2032|2d n , under the condition that \u03bb n d = o(|U \u2032|). We now need to compute ||f||2, ||g||2, ||F||2, ||G||2, ||F \u2032||2, ||G\u2032||2 to apply Lemma 2.2. We will be slightly pedantic and write out more details for the estimate of ||f||2, and let the reader \ufb01ll in the 7 \fvery similar details that we omit from the sequel. ||f||2 2 = X x,y\u2208U\u2032 f(x, y)f(x, y) = X x,y\u2208U\u2032 |{(x, y, u) \u2208(U \u2032)3 : x \u223cy, y \u223cu}|2 = X x,y\u2208U\u2032 |{(x, y, u) \u2208V 3 : x \u223cy, y \u223cu}| \u00b7 |{(x, y, u\u2032) \u2208V : x \u223cy, y \u223cu\u2032}|. So for an adjacent pair x, y \u2208U \u2032, the contribution to ||f||2 2 is the number of pairs of vertices u and u\u2032 in V such that y is adjacent to both. In the case that u and u\u2032 are distinct, we get a 3-star. In the (degenerate) case that u = u\u2032, we get a path of length 2. Note that if x and y are not adjacent, they will not contribute to the sum. So ||f||2 2 is the sum of the number of 3-stars in G plus the number of 2-paths in G. We now set out to estimate these quantities. We already saw that Proposition 2.3, that P2(U \u2032) = (1 + o(1))|U \u2032|3d2n\u22122. So it only remains to estimate the number of 3-stars in U \u2032 and verify that this is much more than P2(U \u2032). Using the fact that each vertex in U \u2032 has at most 2|U|d/n neighbors in U, we can conclude that the number of 3-stars is O \u0000|U|4d3n\u22123\u0001 , and this clearly dominates our estimate of P2(U \u2032), as \u03bbn/d = o(U) by assumption, and \u03bb \u22651 (by considering the trace of A2 one can see that \u03bb \u2273 \u221a d). Putting everything together gives us the estimate ||f||2 2 \u226a|U|4d3 n3 . Moreover, by analogous arguments, neglecting similarly degenerate terms, one has ||g||2 2 = ||g||2 1 = (1 + o(1))|U \u2032|2d n . Similarly, ||F||2 2 is bounded by the number of paths of length 4, and ||F \u2032||2 2 is bounded by the number of stars with 4 edges, i.e. ||F||2 2, ||F \u2032||2 2 \u226a|U|5d4 n4 . We now bound ||G\u2032||2 2 and ||G||2 2. It is clear that these are at most the number of paths of length 2, and so we have ||G\u2032||2 2, ||G||2 2 \u226a|U|3d2/n2. Putting these bounds to Proposition 2.2 and noting that X x\u223cz,y\u223cw f(x, y)g(z, w) = H1(U \u2032) gives us \f \f \f \fH1(U \u2032) \u2212|U \u2032|5d5 n5 \f \f \f \f \u226a\u03bb2 |U \u2032|3d2 n2 + d\u03bb n |U \u2032|4d3 n3 . This completes the proof. Theorem 3.2. Let G be an (n, d, \u03bb)-graph and U a subset of vertices such that \u03bbn/d = o(|U|). 8 \fThen there is a subset U \u2032 \u2282U with |U \u2032| = (1 \u2212o(1))|U| such that \f \f \f \fH2(U \u2032) \u2212(1 + o(1))|U \u2032|5d6 n6 \f \f \f \f \u226a\u03bb2 \u00b7 |U \u2032|3d2 n2 . Proof. By Lemma 2.5, we may choose a subset U \u2032 \u2282U with |U \u2032| = (1 \u2212o(1))|U| such that no vertex in U \u2032 has degree more than 2|U|d n in U. As before, we will show that this subset satis\ufb01es the conclusion. We recall that H2(U \u2032) counts the number of 5-tuples (x, y, z, u, v) \u2208(U \u2032)5 such that x \u223cu, u \u223cz, z \u223cy, x \u223cy, u \u223cv, v \u223cy. That is, it is the number of (labeled, possibly degenerate) K2,3 in U \u2032. For u, y, z \u2208U \u2032, de\ufb01ne fu(y) as the number of cycles of length 4 with vertices in U \u2032 such that u and y form a diagonal, de\ufb01ne gu(z) = 1 if u \u223cz and 0 otherwise. Summing over all u in U \u2032 and y \u223cz, we have, H2(U \u2032) = X u\u2208U\u2032 X y\u223cz fu(y)gu(z) = X u\u2208U\u2032 \u27e8fu, Agu\u27e9. Applying Lemma 2.1 for each choice of u and summing gives us \f \f \f \f \f \f H2(U \u2032) \u2212d n X u X y\u2208U\u2032 fu(y) X z\u2208U\u2032 gu(z) \f \f \f \f \f \f \u226a\u03bb X u\u2208U\u2032  X y fu(y)2 !1/2  X z gu(z)2 !1/2 . Notice that X u,y,z\u2208U\u2032 fu(y)gu(z) = |H1(U \u2032)|. By the Cauchy-Schwarz inequality, one has X u\u2208U\u2032  X y fu(y)2 !1/2  X z gu(z)2 !1/2 \u2264  X u,y fu(y)2 !1/2 \u00b7  X u,z gu(z)2 !1/2 . Now P fu(y)2 is the number of (labeled, possibly degenerate) copies of K2,4. This can be bounded above by the number of K2,3 times the degree of one of the vertices in the part of size 2 (note that this is on average giving up a factor of about d n). Using the fact that each vertex in U \u2032 has at most 2|U|d/n neighbors in U, we have X u X y fu(y)2 \u2264|H2(U \u2032)| \u00b7 2|U|d n . Moreover, the sum P u,z gu(z)2 equals P1(U \u2032), so it is (1 + o(1))|U \u2032|2d/n by Lemma 2.1. Putting these computations together and using Theorem 3.1, one has \f \f \f \fH2(U \u2032) \u2212(1 + o(1))|U \u2032|5d6 n6 \f \f \f \f \u226a\u03bb2 \u00b7 |U \u2032|3d2 n2 . This completes the proof. 9 \fRemark 3.1. The above two lemmas say that when \u03bb(n/d)3/2 = o(|U|), there is a subset U \u2032 \u2282U with |U \u2032| \u226b|U| such that H1(U \u2032) = (1 + o(1))|U \u2032|5d5 n5 , H2(U \u2032) = o(H1(U \u2032)). In the proof of Theorem 3.2, if we assume that any two vertices in U have at most \u03b3 common neighbors, and \u039b = min{\u03b3, 2|U|d/n}, then X u,y fu(y)2 \u2264H2(U \u2032) \u00b7 \u039b. This implies the following strengthened version. Theorem 3.3. Let G be an (n, d, \u03bb)-graph and U a subset of vertices such that \u03bbn/d = o(|U|). If any two vertices in U have at most \u03b3 common neighbors, and \u039b = min{\u03b3, 2|U|d/n}, then there is a subset U \u2032 \u2282U with |U \u2032| = (1 \u2212o(1))|U| such that \f \f \f \fH2(U \u2032) \u2212(1 + o(1))|U \u2032|5d6 n6 \f \f \f \f \u226a\u03bb2 \u00b7 |U \u2032|2d n \u00b7 \u039b. Next we give estimates on H+ 3 , the supergraph of H3 with respect to the number of copies of H3, and \ufb01nally we give an upper bound on H\u2212 3 . We were not able to give an asymptotic formula for H3 and we leave this as an interesting open problem. Later in the proof of Theorem 1.5 we will give only a lower bound on the number of H3 and this will be enough for our purposes. In order to estimate H+ 3 we will \ufb01rst need to estimate one more con\ufb01guration. De\ufb01ne H4(U \u2032) to be the number of 6-tuples (y, z, u, v, u\u2032, x) in (U \u2032)6 such that y \u223cz, z \u223cu, u \u223cv, v \u223cu\u2032, u\u2032 \u223cx, x \u223cy, z \u223cu\u2032. Theorem 3.4. Let G be an (n, d, \u03bb)-graph and U a subset of vertices such that |U| \u226b\u03bb(n/d)3/2. Then there is a subset U \u2032 \u2282U with |U \u2032| = (1 \u2212o(1))|U| such that H4(U \u2032) \u226a|U \u2032|6d7 n7 . Proof. By Theorem 2.4, we have that the number of C4 in U is O \u0012|U|4d4 n4 \u0013 , since |U| \u226b\u03bb(n/d)3/2. Let L be the set of vertices in U which are contained in at least C|U|3d4/n4 copies of C4 in U, and de\ufb01ne U \u2032 = U \\ L. Then for any \u03f5 > 0, there is a C such that |U \u2032| \u2265(1 \u2212\u03f5)|U| and every vertex in U \u2032 has at most C|U|3d4/n4 copies of C4 containing it. First we de\ufb01ne f(z, u\u2032) = #{(u, v) \u2208U \u2032 \u00d7 U \u2032 : z \u223cu, u \u223cv, v \u223cu\u2032, u\u2032 \u223cz}, and g(y, x) = 1 if y \u223cx and 0 otherwise. 10 \fThe strategy is to apply Proposition 2.2. It is clear that P z,u\u2032\u2208U\u2032 f(z, u\u2032) counts the number of 4-cycles in U \u2032, so we have that X z,u\u2032\u2208U\u2032 f(z, u\u2032) = O \u0012|U \u2032|4d4 n4 \u0013 . Since \u03bb(n/d) = o(|U|), we have P y,z\u2208U\u2032 g(y, z) = (1 + o(1))|U \u2032|2d/n by Proposition 2.3. On the other hand, P z,u\u2032\u2208U\u2032 f(z, u\u2032)2 is at most H4(U \u2032), and P y,x\u2208U\u2032 g(y, x)2 = P y,x\u2208U\u2032 g(y, x) \u226a |U \u2032|2d/n. In the setting of Proposition 2.2, we can check directly that P z F(z)2 counts the number of pairs of 4-cycles sharing a common point, so is bounded by O \u0012|U \u2032|7d8 n8 \u0013 , by the de\ufb01nition of U \u2032. Moreover, P y G(y)2 is at most the number of paths of length 2 in U \u2032, which is at most O \u0000|U \u2032|3d2/n2\u0001 by Proposition 2.3. The same estimates hold for P u\u2032 F(u\u2032)2 and P x G(x)2. Putting these estimates together and noting that H4(U \u2032) = P x\u223cu\u2032,y\u223cz f(x, y)g(z, u\u2032) gives us by Proposition 2.2 that H4(U \u2032) \u226a|U \u2032|6d7 n7 + \u03bb2H4(U \u2032)1/2 \u0012|U \u2032|2d n \u00131/2 + \u03bb|U \u2032|5d6 n6 . Thus, H4(U \u2032) \u226a|U \u2032|6d7 n7 + \u03bb4 |U \u2032|2d n \u226a|U \u2032|6d7 n7 . With this in hand, we are able to estimate H+ 3 Theorem 3.5. Let G be an (n, d, \u03bb)-graph and U a subset of vertices such that |U| \u226b\u03bb(n/d)3/2. Then there is a subset U \u2032 \u2282U with |U \u2032| = (1 \u2212o(1))|U| such that \f \f \f \fH+ 3 (U \u2032) \u2212H3(U \u2032)d n \f \f \f \f \u226a\u03bb \u0012 H2(U \u2032)|U \u2032|d n \u00131/2 \u0012|U \u2032|6d8 n8 + \u03bbH2(U \u2032) \u00131/2 . Proof. As before, by Lemma 2.5, we pass to a subset U \u2032 \u2282U with |U \u2032| = (1 \u2212o(1))|U| such that no vertex in U \u2032 has degree more than 2|U|d n in U. We furthermore may assume that the conclusion of Theorem 3.4 holds on U \u2032. We recall that H+ 3 (U \u2032) counts the number of 7-tuples (x, y, z, u, v, u\u2032, x\u2032) \u2208(U \u2032)7 such that x \u223cy, y \u223cz, z \u223cu, u \u223cx, u \u223cv, v \u223cu\u2032, u\u2032 \u223cz, u\u2032 \u223cx\u2032, x\u2032 \u223cy, x \u223cu\u2032. To prove this theorem, we proceed as follows. Given y, z, v \u2208U \u2032 such that y \u223cz, we de\ufb01ne fy,z,v(u\u2032) := #{x\u2032 \u2208U \u2032 : x\u2032 \u223cy, u\u2032 \u223cz, u\u2032 \u223cv, u\u2032 \u223cx\u2032}, and gy,z,v(x) := #{u \u2208U \u2032 : x \u223cy, x \u223cu, u \u223cv, u \u223cz}. 11 \fNote that fy,z,v(u\u2032) is the number of vertices in N(u\u2032) \u2229N(y) times the indicator that u\u2032 \u2208N(z) \u2229 N(v) and gy,z,v(x) is the number of vertices in N(v) \u2229N(x) \u2229N(z) times the indicator that x \u2208N(y). Then we have that H+ 3 (U \u2032) = X y,z,v,y\u223cz X x\u223cu\u2032 fy,z,v(u\u2032)gy,z,v(x) = X y,z,v,y\u223cz \u27e8f, Ag\u27e9. We \ufb01rst observe that X y,z,v,y\u223cz ||fy,z,v||1||gy,z,v||1 = H3(U \u2032). Next, let y \u223cz and v be \ufb01xed. For each u\u2032, we have that (fy,z,v(u\u2032))2 is at most the number of homomorphisms to K2,3 where y and u\u2032 are in the part of size 2 and z is in the part of size 3, times the indicator that u\u2032 \u223cv. Therefore, this is at most the number of K2,3 with y, u\u2032 in the part of size 2 times the degree of u\u2032. This implies X y,z,v,y\u223cz ||fy,z,v||2 2 \u226aH2(U \u2032) \u00b7 |U|d n . In the next step, we bound the sum P y,z,v,y\u223cz ||gy,z,v||2 2 which is much more complicated. We denote this sum by M. We have M counts the number of 6-tuples (y, z, u, v, u\u2032, x) \u2208U \u2032 such that y \u223cz, z \u223cu, u \u223cv, v \u223cu\u2032, u\u2032 \u223cx, x \u223cy, x \u223cu, z \u223cu\u2032. To bound M from above, we proceed as follows. More precisely, given z \u223cu\u2032, de\ufb01ne kz,u\u2032(u) = #{v \u2208U \u2032 : v \u223cu, v \u223cu\u2032, z \u223cu}, and hz,u\u2032(x) = #{y \u2208U \u2032 : y \u223cx, y \u223cz, x \u223cu\u2032}. Then it is clear that M = X x,u\u2208U\u2032 x\u223cu X z,u\u2032\u2208U\u2032 z\u223cu\u2032 kz,u\u2032(u)hz,u\u2032(x). To apply Lemma 2.1, the following estimates are needed. X z,u\u2032\u2208U\u2032 z\u223cu\u2032 |kz,u\u2032(u)|2 = H2(U \u2032), X z,u\u2032\u2208U\u2032 z\u223cu\u2032 |hz,u\u2032(x)|2 = H2(U \u2032). We also need to bound P z,u\u2032 kz,u\u2032(u)hz,u\u2032(x). This sum counts the number of 6-tuples (y, z, u, v, u\u2032, x) \u2208 U \u2032 such that y \u223cz, z \u223cu, u \u223cv, v \u223cu\u2032, u\u2032 \u223cx, x \u223cy, z \u223cu\u2032. Note that this is exactly H4(U \u2032). By Theorem 3.4, we have that H4(U \u2032) \u226a|U \u2032|6d7 n7 , 12 \fsince |U \u2032| \u226b\u03bb(n/d)3/2. Applying Lemma 2.1, we have M = X z\u223cu\u2032 \u27e8kz,u\u2032, Ahz,u\u2032\u27e9\u226a|U \u2032|6d8 n8 + \u03bbH2(U \u2032). Hence, the theorem follows from Lemma 2.1. Theorem 3.6. Let G be an (n, d, \u03bb)-graph and U a subset of vertices such that n\u03bb/d = o(|U|), then there is a subset U \u2032 \u2282U of size (1 \u2212o(1))|U| such that H\u2212 3 (U \u2032) \u226a|U \u2032|6d7 n7 + \u03bb2 |U \u2032|4d4 n4 + \u03bb|U \u2032|5d5 n5 + \u03bb2 |U \u2032|4d7/2 n7/2 + \u03bb3 |U \u2032|3d5/2 n5/2 + \u03bb4 |U \u2032|2d n . Proof. As before, pass to a subset U \u2032 of size (1 \u2212o(1))|U \u2032| where each vertex in U \u2032 has at most 2|U|d n neighbors in U. Fix z \u2208U \u2032. De\ufb01ne fz(x) to be the number of tuples (z, x, v, u) \u2208U \u20324 such that z \u223cx, x \u223cv, v \u223c u, u \u223cz, and gz(x\u2032) to be the number of tuples (z, x\u2032, y) \u2208U \u20323 such that z \u223cy, y \u223cx\u2032. Then it is clear that H\u2212 3 (U \u2032) \u2264 X z\u2208U X x\u223cx\u2032 fz(x)gz(x\u2032). We \ufb01rst note that the sum P z ||fz||1||gz||1 is equal to the number of tuples (z, x, v, u, y, x\u2032) such that z \u223cx, x \u223cv, v \u223cu, u \u223cz, z \u223cy, y \u223cx\u2032. This number is at most H1(U \u2032) \u00b7 2|U|d/n. To bound H\u2212 3 (U \u2032) from above, we will apply Lemma 2.1. To proceed further, we need to bound the L2-norm of fz and gz. In the way we de\ufb01ne the function fz and gz, it is not hard to see that the sum P z ||fz||2 2 is at most the number of 6-cycles C6(U \u2032), which by Theorem 2.4 is bounded from above by (1 + o(1))|U \u2032|6d6 n6 + O \u0012\u03bb4|U \u2032|2d n \u0013 . Similarly, we also have the sum P z ||gz||2 2 is at most the number of 4-cycles C4(U \u2032), which is bounded from above by (1 + o(1))|U \u2032|4d4 n4 + O \u0012\u03bb2|U \u2032|2d n \u0013 . In other words, we can conclude that H\u2212 3 (U \u2032) \u2264H1(U \u2032)|U \u2032|d2 n2 + \u03bb|U \u2032|5d5 n5 + \u03bb2 |U \u2032|4d7/2 n7/2 + \u03bb3 |U \u2032|3d5/2 n5/2 + \u03bb4 |U \u2032|2d n . Applying Theorem 3.1 completes the proof. Remark 3.2. In practice, con\ufb01gurations in Theorems 3.5 and 3.6 might not exist. 4 Proof of Theorem 1.4 To prove Theorem 1.4, we need to show that there exists a set of two distinct points, say x1, x2 \u2208U, such that the restriction of H to U yields every possible function from {x1, x2} to {0, 1}. This is 13 \fequivalent to saying that there exists u\u2217, u1, u2, u12 \u2208U such that x1 \u223cu12, x2 \u223cu12, x1 \u223cu1, x2 \u0338\u223cu1, x1 \u0338\u223cu2, x2 \u223cu2, and x1 \u0338\u223cu\u2217, x2 \u0338\u223cu\u2217. We \ufb01rst prove the existence of u1, u2, u12 which are joined to x1 and x2 in the prescribed way. By Lemma 2.5 and Theorem 3.1, we may choose a subset U \u2032 \u2282U with |U \u2032| = (1 \u2212o(1))|U| such that no vertex in U \u2032 has degree more than 2|U|d n in U and that satis\ufb01es \f \f \f \fH1(U \u2032) \u2212|U \u2032|5d5 n5 \f \f \f \f \u226a\u03bb2 |U \u2032|3d2 n2 + \u03bb|U \u2032|4d4 n4 . We will show that x1, x2, u1, u2, u12 may be found in U \u2032. Afterwards, we will show that by the de\ufb01nition of U \u2032 there will be a vertex u\u2217\u2208U which is adjacent to neither x1 nor x2. Hence it su\ufb03ces to show the existence of these 5 vertices in U \u2032 with the prescribed edges and non-edges. We do this by \ufb01nding a path on 5 vertices u1 \u223cx1 \u223cu12 \u223cx2 \u223cu2. Such a path will satisfy our requirements if and only if we have u1 \u0338\u223cx2 and u2 \u0338\u223cx1. In order to guarantee the existence of such a path we count all paths and then count paths where at least one of the forbidden edges is present. We know from Proposition 2.3 that the number of paths on 5 vertices in U \u2032 with it at least (1 \u2212o(1))|U \u2032|5d4 n4 , (1) because of the restriction that \u03bbn d = o(|U|). Note that there are also this many paths where all 5 vertices are distinct, again by Proposition 2.3. Such a path will satisfy our requirements unless x1 \u223cu2 or x2 \u223cu1. If either of these forbidden edges occurs, we have found con\ufb01guration H1. By the de\ufb01nition of U \u2032 we have that H1(U \u2032) \u2264|U \u2032|5d5 n5 + O \u0012 \u03bb2 |U \u2032|3d2 n2 + \u03bb|U \u2032|4d4 n4 \u0013 , This quantity is much smaller than the number of paths in (1) by the hypothesis on the size of U. Choosing any vertex u\u2217\u2208U which is not in N(x1) \u222aN(x2) completes the proof. We are guaranteed that such a vertex exists, as x1, x2 \u2208U \u2032, so neither has more than 2|U|d n neighbors in U \u2032. So |N(x1) \u222aN(x2)| \u22644|U|d n , leaving at least |U \u2032| \u22124|U|d n choices for u\u2217in U \u2032 \\ (N(x1) \u222aN(x2)). The number of choices of u\u2217is then strictly positive by our assumptions. 5 Proof of Theorem 1.5 The proof of Theorem 1.5 is a bit involved, so we \ufb01rst give a sketch. 14 \f5.1 Sketch of proof To prove the VC-dimension is at least 3, we need to \ufb01nd vertices x1, x2, x3, y1, y2, y3, y12, y13, y23, y13, y\u2217\u2208 U with x1, x2, x3 distinct such that 1. x1 \u223cy123, x2 \u223cy123, x3 \u223cy123 2. x1 \u223cy12, x2 \u223cy12, x3 \u0338\u223cy12, and similarly for y13 and y23. 3. x1 \u223cy1, x2 \u0338\u223cy1, x3 \u0338\u223cy1, and similarly for y2 and y3. 4. x1 \u0338\u223cy\u2217, x2 \u0338\u223cy\u2217, x3 \u0338\u223cy\u2217. If we can \ufb01nd vertices in such a con\ufb01guration, then the set {hx1, hx2, hx3} is shattered by H(U) and this shows that the VC-dimension of U is at least 3. We again note that for the de\ufb01nition of shattering, it does not matter what adjacency or non-adjacency we see between xi and xj or between any of the y vertices. We also note that for the de\ufb01nition of shattering to be satis\ufb01ed, it is possible that some y and some xj are the same vertex. However, we will avoid this situation because the possible presence of loops in our graphs make it hard to analyze. To prove that we can \ufb01nd a set of vertices with the above properties, we use the count of con\ufb01gurations in Section 3. To prove Theorem 1.5, we do the following steps. Step 1: We \ufb01rst pass to a subset U \u2032 where we may count the con\ufb01gurations from Section 3. Then we argue that it is enough to \ufb01nd vertices x1, x2, x3, y123, y12, y23, y13 in U \u2032. Step 2: Next we count the number of con\ufb01gurations H1 and use Cauchy-Schwarz to lower bound the number of tuples satisfying \u2022 x1 \u223cy123, x2 \u223cy123, x3 \u223cy123 \u2022 x1 \u223cy12, x2 \u223cy12 \u2022 x1 \u223cy13, x3 \u223cy13 \u2022 x2 \u223cy23, x3 \u223cy23 \u2022 x2 \u0338\u223cy13 \u2022 x1 \u0338= x2, x3 \u0338= x2 \u2022 y123 \u0338= y12, y123 \u0338= y23 Step 3: In the previous step, some of the vertices may overlap. Since we must have that x1 \u0338= x3, next we show that the number of con\ufb01gurations in Step 2 with x1 = x3 is much smaller than the total. Step 4: After the step 3, we get a lower bound on the number of tuples satisfying \u2022 x1 \u223cy123, x2 \u223cy123, x3 \u223cy123 \u2022 x1 \u223cy12, x2 \u223cy12 \u2022 x1 \u223cy13, x3 \u223cy13 \u2022 x2 \u223cy23, x3 \u223cy23 15 \f\u2022 x2 \u0338\u223cy13 \u2022 x1 \u0338= x2, x3 \u0338= x2, x1 \u0338= x3 \u2022 y123 \u0338= y12, y123 \u0338= y23 Step 5: For those tuples in the Step 4, we need to remove tuples with y12 \u223cx3 and y23 \u223cx1 and tuples with x2 = y12 or x3 = y12. If there is an adjacency we have the con\ufb01guration Hxu\u2032 3 and if there is an overlap of vertices we have the con\ufb01guration H\u2212 3 . Step 6: Using the assumption in the statement of the theorem, we conclude the proof. 5.2 Details Proof of Theorem 1.5. First we pass to a subset U \u2032 where our con\ufb01gurations are well-behaved. By Lemma 2.5, we may consider a subset U \u2032 \u2282U such that all vertices in U \u2032 have between 1 2 |U|d n and 2|U|d n neighbors in U. By Remark 3.1 and Theorem 3.5, we may assume that H1(U \u2032) is close to its expected value, that H2(U \u2032) = o(H1(U \u2032)), and that H+ 3 (U \u2032) \u2264\u03f5(|U|7d9/n9 + H3(U \u2032)) where \u03f5 can be taken arbitrarily small by choosing C large enough in the hypotheses of the theorem. We focus on the existence of the vertices x1, x2, x3, y123, y12, y23, y13 in U \u2032. If we can \ufb01nd such a subcon\ufb01guration, then by the assumption in the hypotheses of the theorem, we may \ufb01nd vertices y1, y2, y3. Since |U| is much larger than |U|d n and the vertices in U \u2032 have at most 2 |U|d n neighbors in U, we may also \ufb01nd the vertex y\u2217which completes the con\ufb01guration, as in the \ufb01nal step of the proof of Theorem 1.4. Therefore it su\ufb03ces to \ufb01nd the vertices x1, x2, x3, y123, y12, y23, y13 in U \u2032 satisfying the adjacency and non-adjacency conditions. Note that this is exactly con\ufb01guration H3. Since each vertex in U \u2032 has at most 2|U|d/n neighbors, the number of 3-stars is at most 8|U|4d3/n3. Under the condition \u03bbn/d = o(|U|), we also know from Proposition 2.3 that the number of 3-paths is (1 + o(1))|U|4d3/n3 and from Theorem 2.4 that C4(U) = (1 + o(1)) |U|4d4 n4 . For any {x, y, z, u, v} with x \u223cy, y \u223cz, z \u223cu, u \u223cx, u \u223cv (that is, that form con\ufb01guration H1), if x = z we have a P3, if y = u we have a star on 3 edges, if v \u2208{x, y, z} we have a 4-cycle, and if y \u223cv we have con\ufb01guration H2. So, it follows from Remark 3.1 that the number of tuples in U \u2032 satisfying x \u223cy, y \u223cz, z \u223cu, u \u223cx, u \u223cv, y \u0338\u223cv, x \u0338= z, y \u0338= u, v \u0338\u2208{x, y, z} (2) is at least (1 \u2212\u03f5)|H1| as long as |U \u2032| \u2265C\u03bb(n/d)3/2, for any \u03f5 and a large enough constant C. We denote the set of those tuples by H\u2032 1. We will de\ufb01ne the notation H\u2032 1(x, y, z, u, v) as the indicator that x, y, z, u, v satisfy (2), that is that they form a con\ufb01guration in H\u2032 1. De\ufb01ne f(y, z, v) = X x,u\u2208U\u2032 H\u2032 1(x, y, z, u, v). Then we have that X y,z,v, y\u223cz (f(y, z, v))2 = H3(U \u2032). We will complete the proof by bounding this number from below and then showing that the number of these homomorphisms which are either not injective or which have an edge that is forbidden is 16 \fsmaller than the total number. Proceeding with this, we have |U|10d10 n10 \u226a|H\u2032 1|2 = \uf8eb \uf8ed X y,z,v\u2208U, y\u223cz f(y, z, v) \uf8f6 \uf8f8 2 \u2264 X y,z,v f(y, z, v)2 \u00b7 X y,z,v 1y\u223cz. Notice that X y,z,v 1y\u223cz \u226a|U \u2032| \u00b7 |U \u2032|d n \u00b7 |U \u2032|, by the degree restriction on U \u2032. This means that X y,z,v f(y, z, v)2 \u226b|U \u2032|7d9 n9 . Given a \ufb01xed y \u223cz and v, the quantity (f(y, z, v))2 counts the number of 4-tuples (x, u, x\u2032, u\u2032) so that H\u2032 1(x, y, z, u, v) = H\u2032 1(x\u2032, y, z, u\u2032, v) = 1. In Figure 1, we think of y, z, v as playing the roles of x1, y123, y23 respectively, and we hope for a 4-tuple (x, u, x\u2032, u\u2032) that play the roles of (y13, x3, y12, x2). By the de\ufb01nition of H\u2032 1, we have that each {x, y, z, u, v} forming a copy of H\u2032 1 satis\ufb01es x \u0338= z, y \u0338= u, v \u0338\u2208{x, y, z}, and y \u0338\u223cv. This means that the number of tuples (x1, x2, x3, y12, y13, y23, y123) \u2208(U \u2032)7 such that \u2022 x1 \u223cy123, x2 \u223cy123, x3 \u223cy123 \u2022 x1 \u223cy12, x2 \u223cy12 \u2022 x1 \u223cy13, x3 \u223cy13 \u2022 x2 \u223cy23, x3 \u223cy23 \u2022 x1 \u0338= x2, x1 \u0338= x3 \u2022 y123 \u0338= y12, y123 \u0338= y13, y123 \u0338= y23. \u2022 y23 \u0338\u2208{y13, y12, x1, x2} \u2022 x1 \u0338\u223cy23 is at least \u2126 \u0000|U|7d9/n9\u0001 . Such a 7-tuple satis\ufb01es all of our conditions unless one of the following occurs. \u2022 y12 = y13 \u2022 x2 = x3 \u2022 x2 = y13 \u2022 x3 = y12 \u2022 x2 \u223cy13 \u2022 x3 \u223cy12 If x2 \u223cy13 or x3 \u223cy12, we get a con\ufb01guration isomorphic to H+ 3 . Hence by Theorem 3.5, the number of these is at most \u03f5(|H3(U \u2032)|) whenever |U \u2032| \u2265C\u03bb(n/d)13/7, (3) 17 \ffor any \u03f5 and a large enough C. If x2 = x3, then the vertices {x1, x2, y12, y13, y123} form con\ufb01guration H2 and furthermore y23 \u223cx2. Thus the number of 6-tuples of this form is bounded above by |H2(U \u2032)| \u00b7 2|U|d n . By Theorem 3.2, this is at most O \u0012|U|6d7 n7 + \u03bb2 |U|4d3 n3 \u0013 . This is smaller than \u03f5|U \u2032|7d9/n9 when |U| \u2265C\u03bb2/3(n/d)2, (4) for a large enough absolute constant C. Similarly, if y12 = y13, then the vertices {y12, y123, x1, x2, x3} form a H2 and additional y23 is adjacent to x2 (and x3). Hence the number of 6-tuples of this form is also bounded above by O \u0012|U|6d7 n7 + \u03bb2 |U|4d3 n3 \u0013 . Finally, if x2 = y13 or equivalently if x3 = y12, then we have con\ufb01guration H\u2212 3 . By Theorem 3.6, the number of these is at most O   |U \u2032|6d7 n7 + \u03bb2 |U \u2032|4d4 n4 + \u03bb|U \u2032|5d5 n5 + \u03bb2 |U \u2032|4d7/2 n7/2 + \u03bb3 |U \u2032|3d5/2 n5/2 + \u03bb4 |U \u2032|2d n ! , which is smaller than |U \u2032|7d9/n9 when |U| \u2265C max{\u03bb2/3(n/d)2, \u03bb(n/d)3/2, \u03bb3/4(n/d)13/8, \u03bb4/5(n/d)8/5} \u2265C max n \u03bb2/3(n/d)2, \u03bb(n/d)13/7o . (5) Therefore if C is chosen to be a large enough absolute constant, we have that there is a 7-tuple such that none of the forbidden adjacencies or vertex identi\ufb01cations occur, and therefore the VCdimension of H(U) is at least 3 under |U| \u2265C max n \u03bb2/3(n/d)2, \u03bb(n/d)13/7o . 6 Proofs of Theorems 1.7, 1.8, and 1.9 The proofs of Theorems 1.7 and 1.9 are almost the same, so we only present one of them. Proof of Theorem 1.7. The only detail we need to check is the \ufb01rst condition, namely, given any three vertices v1, v2, v3 in U, we can \ufb01nd three vertices u1, u2, u3 in U such that ui \u00b7 vj = 1 if and only if i = j. Using the fact that any two distinct hyperplanes in Ft q intersect in at most qt\u22122 points. This means that |N(vi) \u2229N(vj)| \u2264qt\u22122. On the other hand, by Lemma 2.5 we may pass to a subset U \u2032 such that for each i, |N(vi) \u2229U \u2032| = \u0398(q\u22121|U|). So, the condition |U| \u2265Cqt\u22121 is enough to guarantee that we will \ufb01nd such vertices v1, v2, v3 as long as C is chosen large enough. Proof of Theorem 1.8. In this particular setting, we can use the geometric properties to improve 18 \fthe argument in the proof of Theorem 1.5. We will run the same proof as in Theorem 1.5 except we also remove all unit vectors from U when de\ufb01ning U \u2032. Since there are at most O(q2) unit vectors in F3 q, we still have that |U \u2032| = (1 + o(1))|U|. Now we use geometric properties to show that for some of the con\ufb01gurations, we either have better upper bounds or they do not exist. More precisely, when estimating copies of con\ufb01guration 2, the fact that the intersection of two distinct planes is either a line or a null set, we have that any two vertices in the graph have at most q common neighbors. We may therefore use Theorem 3.3 and this implies that the condition (4) is replaced by |U \u2032| \u2265Cq5/2. Next we show that any copy of H\u2212 3 contains a unit vector and hence by the de\ufb01nition of U \u2032 we may ignore the con\ufb01guration. Assume there is a copy of H\u2212 3 with labels as in Figure 1, so x = u\u2032. By the adjacencies, we have that the planes {w : v \u00b7 w = 1}, {w : x \u00b7 w = 1}, and {w : z \u00b7 w = 1} all contain u. Since the intersection of three planes is either a line or is empty, the intersection must be a line. But the line containing u and x is the intersection of the planes {w : v \u00b7 w = 1} and {w : z \u00b7 w = 1}, and hence the line containing x and u is also the intersection of all three planes. But this now implies that x \u00b7 x = 1. Hence, the condition (5) is not needed. With a similar argument and the fact that x1 \u0338\u223cy23, we can check that the con\ufb01guration H+ 3 does not appear (this argument appears at the end of the paper [6]). So the condition (3) is not needed. In conclusion, in this setting, we only need the condition |U \u2032| \u2265Cq5/2 to ensure that the dimension is at least three. To see the upper bound on the VC-dimension in F3 q, notice that no quadruple of points can be shattered, as any three of them determine a plane. Since dot-products are constant on planes, any quadruple of points that have the same non-zero dot product with a point will lie on a plane. Thus any triple of them will lie on the same plane. So if any three of the points from the quadruple determine a given non-zero dot product with a point in F3 q, so will the fourth, and we cannot shatter any set of four points. This completes the proof. 7 Acknowledgements T. Pham would like to thank to the VIASM for the hospitality and for the excellent working conditions. M. Tait was partially supported by National Science Foundation grant DMS-2011553."
+                },
+                {
+                    "url": "http://arxiv.org/abs/2208.04399v1",
+                    "title": "Geometric structures in pseudo-random graphs",
+                    "abstract": "In this paper, we provide a general framework for counting geometric\nstructures in pseudo-random graphs. As applications, our theorems recover and\nimprove several results on the finite field analog of questions originally\nraised in the continuous setting. The results present interactions between\ndiscrete geometry, geometric measure theory, and graph theory.",
+                    "authors": "Thang Pham, Steven Senger, Michael Tait, Vu Thi Huong Thu",
+                    "published": "2022-08-08",
+                    "updated": "2022-08-08",
+                    "primary_cat": "math.CO",
+                    "cats": [
+                        "math.CO"
+                    ],
+                    "main_content": "Introduction 2 1.1 Cartesian product structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Distribution of cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Distribution of disjoint trees . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Proof of Theorem 1.3 7 2.1 Square-norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 A weak hypergraph regularity lemma . . . . . . . . . . . . . . . . . . . . . . 9 2.3 A generalized von-Neumann type estimate . . . . . . . . . . . . . . . . . . . 10 3 Proof of Theorem 1.5 15 3.1 The \ufb01rst counting lemma for cycles . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 The second counting lemma for cycles . . . . . . . . . . . . . . . . . . . . . . 19 3.3 Distribution of paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.4 Proof of Theorem 1.5 using the \ufb01rst counting lemma . . . . . . . . . . . . . 22 3.5 Proof of Theorem 1.5 using the second counting lemma . . . . . . . . . . . . 26 \u2217University of Science, Vietnam National University, Hanoi. Email: thangpham.math@vnu.edu.vn \u2020Department of Mathematics, Missouri State University. Email: StevenSenger@MissouriState.edu \u2021Department of Mathematics & Statistics, Villanova University. Email: michael.tait@villanova.edu \u00a7University of Science, Vietnam National University, Hanoi. Email: VuThiHuongThu T64@hus.edu.vn 1 arXiv:2208.04399v1  [math.CO]  8 Aug 2022 \f4 Proofs of Theorem 1.7 and Theorem 1.8 28 4.1 Technical lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.2 Proof of Theorem 1.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.3 Proof of Theorem 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5 Acknowledgements 32 1 Introduction Let Fq be a \ufb01nite \ufb01eld of order q where q is a prime power. The investigation of \ufb01nite \ufb01eld analogs of problems originally raised in geometric measure theory has a long tradition, for instance, the Erd\u02dd os-Falconer distance problem [6, 7, 17], sum-product estimates [4, 10], the Kakeya problem [8, 32], frame theory [15, 16], and restriction problems [14, 22, 23, 27, 28]. Studying these problems over \ufb01nite \ufb01elds is not only interesting by itself, but it also o\ufb00ers new ideas to attack the original questions. Some of these problems can be proved by using results from the in graph theory, for instance, in [17], Iosevich and Rudnev proved the following theorem on the distribution of distances in a given set. Theorem 1.1 (Iosevich-Rudnev, [17]). Let E be a set in Fd q. Assume that |E| \u226bq d+1 2 , then \u2206(E) := {||x \u2212y|| = (x1 \u2212y1)2 + \u00b7 \u00b7 \u00b7 + (xd \u2212yd)2 : x, y \u2208E} = Fq. It is well-known that this theorem can be reproved by using the famous expander mixing lemma. More precisely, the expander mixing lemma states that for an (n, d, \u03bb)-graph G, i.e. a regular graph with n vertices, of degree d, and all other eigenvalues bounded in absolute value by \u03bb, the number of edges in a given vertex set U, denoted by e(U), is bounded from both above and below by the inequality \f \f \f \fe(U) \u2212d|U|2 2n \f \f \f \f \u2264\u03bb|U|. To derive Theorem 1.1 from this estimate, for \u03b1 \u2208F\u2217 q, one just needs to de\ufb01ne the distance graph DG\u03b1 with the vertex set Fd q and there is an edge between two vertices x and y if and only if ||x \u2212y|| = \u03b1. It is not hard to check that DG\u03b1 is a regular Cayley graph with qd vertices, of degree \u223cqd\u22121, and the second eigenvalue is bounded by q d\u22121 2 by using Kloosterman sums [18, 31]. So, for any \u03b1 \u0338= 0, the expander mixing lemma implies directly that any vertex set U of size at least 2q d+1 2 spans at least one edge. We observe that the argument above only made use of the pseudo-randomness properties of the graph, and once the eigenvalues were calculated ignored anything about Fq or the distance function. Because of this observation, this machinery provides a uni\ufb01ed proof for a series of similar questions, for example, one can replace the distance function by bilinear 2 \fforms [11], Minkowski distance function [12], or other functions [30]. From this observation, it is very natural to ask what kind of \ufb01nite \ufb01eld models can be extended to the graph setting? That is, what \u201cgeometric structures\u201d can we guarantee in a general graph with some pseudo-randomness condition? The main purpose of this paper is to provide three such con\ufb01gurations, and the three topics we present here can be viewed as generalizations of the Erd\u02dd os-Falconer distance conjectures, which have been studied intensively in the literature. Our theorems imply several results found previously as special cases. Moreover, as they rely on the pseudo-randomness of an underlying graph, they can be applied in a straightforward manner to other settings, such as modules over \ufb01nite rings. Throughout the paper we say that G is an (n, d, \u03bb)-colored graph with color set D if it is a graph edge-colored with |D| colors such that the subgraph of any \ufb01xed color is an (n, d, \u03bb)-graph. 1.1 Cartesian product structures We \ufb01rst start with the following question about \ufb01nding rectangles in Fd q. Question 1.2. Let E be a set in Fd q and \u03bb, \u03b2 \u2208F\u2217 q. How large does E need to be to guarantee that there are four points w, x, y, z \u2208E such that they form a rectangle of side lengths \u03b1 and \u03b2, i.e. (w \u2212x) \u00b7 (x \u2212y) = 0, (x \u2212y) \u00b7 (y \u2212z) = 0, (y \u2212z) \u00b7 (z \u2212w) = 0, (z \u2212w)(w \u2212x) = 0, (1) and ||w \u2212x|| = ||y \u2212z|| = \u03b1, ||x \u2212y|| = ||z \u2212w|| = \u03b2. (2) Lyall and Magyar [25] proved that for any \u03b4 \u2208(0, 1), there exists an integer q0 = q0(\u03b4) with the following property: if q \u2265q0 and E \u2282F2d q with |E| \u2265\u03b4q2d, then E contains four points a, b, c, and d satisfying (1) and (2). This is the \ufb01nite \ufb01eld model of a result in the same paper which states that for any given rectangle R in R2d, if S \u2282R2d has positive Banach density, then there exists a threshold \u03bb0 = \u03bb0(S, R) such that S contains an isometric copy of \u03bbR for any \u03bb \u2265\u03bb0. Notice that the result in [25] was actually proved in a more general form, for d-dimensional rectangles, though we state it here for 2-dimensional rectangles. In the \ufb01rst theorem of this paper, we extend this result to a general graph setting. For two graphs G and H, the cartesian product of G and H, denoted by G\u25a1H, is the graph where V (G\u25a1H) = V (G) \u00d7 V (H) and (u1, v1) \u223c(u2, v2) if and only if either u1 = u2 and {v1, v2} \u2208E(H) or v1 = v2 and {u1, u2} \u2208V (G). We use S(x) to denote the indicator function of the set S. 3 \fTheorem 1.3. Let Gi be (ni, di, \u03bbi)-graphs with 1 \u2264i \u22642. Set G = G1\u25a1G2. For any 0 < \u03b4\u2032 < \u03b4 < 1, there exists \u03f5 > 0 such that for any S \u2282V (G1\u25a1G2) with |S| \u2265\u03b4|V (G1\u25a1G2)|, if max n \u03bb1 d1 , \u03bb2 d2 o < \u03f5, then N = X (u1,u2)\u2208E(G1),(v1,v2)\u2208E(G2) S(u1, v1)S(u1, v2)S(u2, v1)S(u2, v2) > \u03b4\u20324n1n2d1d2. Theorem 1.3 recovers the theorem on rectangles in Fd q by Lyall and Magyar (Proposition 2.1 in [25]) as follows. Let G1 and G2 be the graphs each with vertex set Fd q where a \u223cb in G1 if ||a \u2212b|| = \u03b1 and x \u223cy in G2 if ||a \u2212b|| = \u03b2. Then G1 and G2 are graphs with qd vertices, degree asymptotically qd\u22121 and \u03bb \u22642q(d\u22121)/2 (see [1, 21], summarized as Theorem 10.1 in [30]). Note that if ||u1 \u2212u2|| = \u03b1 and ||v1 \u2212v2|| = \u03b2, then letting w = (u1, v1), x = (u1, v2), y = (u2, v1) and z = (u2, v2), we have that w, x, y, and z form a rectangle in F2d q with side lengths \u03b1 and \u03b2. Applying Theorem 1.3 to these speci\ufb01c graphs shows that for q large enough, any subset of F2d q of size at least \u03b4q2d contains \u2126(q4d\u22122) rectangles, giving a quantitative strengthening of Lyall and Magyar\u2019s result. Another application of Theorem 1.3 is on the number of rectangles in F2 q with side lengths in a given multiplicative subgroup of Fq, precisely, given a multiplicative subgroup A of Fq, we de\ufb01ne G1 = G2 being the graph with the vertex set Fq and there is an edge between x and y if x \u2212y \u2208A. This is clear that this is a Cayley graph with q vertices, of degree |A|, and it is also well-known that \u03bb \u2264q1/2 (see [19, (1)] for computations). Applying Theorem 1.3, we recover Theorem 1.1 from [19]. 1.2 Distribution of cycles Our motivation of this section comes from the following question. Question 1.4. Let E be a set in Fd q and m \u22654 be an integer. How large does E need to be to guarantee that the number of cycles of the form (x1, . . . , xm) with ||xi \u2212xi+1|| = 1 for all 1 \u2264i \u2264m \u22121, and ||xm \u2212x1|| = 1, is close to the expected number |E|mq\u2212m? Iosevich, Jardine, and McDonald [13] proved that the number of cycles of length m, denoted by Cm(E), satis\ufb01es Cm(E) = (1 + o(1))|E|m qm , (3) whenever |E| \u2265 ( q 1 2(d+2\u2212k\u22122 k\u22121 +\u03b4) : m = 2k, even q 1 2(d+2\u22122k\u22123 2k\u22121 +\u03b4) : m = 2k + 1 odd 4 \fwhere 0 < \u03b4 < 1 2 \u0004 m 2 \u00052. In the continuous setting, this is a di\ufb03cult problem, and there are only a few partial results. For instance, as a consequence of a theorem due to Eswarathasan, Iosevich, and Taylor [9], we know that if the Hausdor\ufb00dimension of E, denoted by s, is at least d+1 2 , then we know that the upper Minkowski dimension of the set of cycles in E is at most 2s \u2212m. If we consider the case of paths, then Bennett, Iosevich, and Taylor [3] showed that there exists an open interval I such that for any sequence {ti}m i=1 of elements in I, we always can \ufb01nd paths of length m + 1 with gaps {ti}m i=1 between subsequent elements in E as long as the Hausdor\ufb00dimension of E is greater than d+1 2 . We refer the reader to [24] for the recent study on this problem. It is worth noting that in the discrete setting, results on distribution of paths also play crucial role in proving (3). In the graph setting, we have the following extension. We note here that we are counting any sequence of m vertices (v1, \u00b7 \u00b7 \u00b7 , vm) with vi \u223cvi+1 and v1 \u223cvm as a cycle of length m. That is, we are counting labeled cycles and we include degenerate cycles in the count. One could combine Theorem 1.5 for various values of m and lemmas used to prove it to obtain results about non-degenerate cycles as well, but we do not do this explicitly here. Theorem 1.5. Let G be an (n, d, \u03bb)-graph and U be a vertex set with \u03bb \u00b7 n d = o(|U|). Let Cm(U) denote the number of (labeled, possibly degenerate) cycles of length m with vertices in U. Then we have \f \f \f \fCm(U) \u2212|U|mdm nm \f \f \f \f = O \u0012\u03bb|U|m\u22121dm\u22121 nm\u22121 + \u03bbm\u22122|U|2d n \u0013 , The error term cannot be improved for m = 4. For instance, we de\ufb01ne a graph with the vertex set F2 q where two vertices (a, b) and (c, d) are adjacent if and only if ac + bd = 1. Using the geometric facts in F2 q that any two lines intersect in at most one point and there is only one line passing through two given points, we can see that this graph contains no C4, even though it is a (q2, q, \u221aq) graph (if one includes loops). We also remark that as a corollary of a result due to Alon [20, Theorem 4.10] we know that the number of cycles in U is close to the expected number as long as |U| \u226b\u03bb(n/d)2. This result is of course weaker than Theorem 1.5. We now discuss how Theorem 1.5 implies and improves previous results. In [13], counting results for cycles are proved in both the distance graph and the dot-product graph over Fd q. Formally, let Gdist t and Gprod t be the graphs on vertex set Fd q where u \u223cv in Gdist t if ||u \u2212v|| = t and u \u223cv in Gprod t if u \u00b7 v = t. As each of these graphs are approximately qd\u22121 regular and with second eigenvalue bounded above by 2q(d\u22121)/2, Theorem 1.5 can be 5 \fapplied. In [13], the same quantitative results are proved for both graphs but with di\ufb00erent methods, and the authors write the following: \u201cWe note that in this paper, we obtain the same results for the distance graph and the dotproduct graph. While the techniques are, at least super\ufb01cially, somewhat di\ufb00erent due to the lack of translation invariance in the dot-product setting, it is reasonable to ask whether a general formalism is possible.\u201d Theorem 1.5 answers this question in a strong way, as it may be applied in a much more general setting than just distance or dot-product graphs. Furthermore, Theorem 1.5 implies the estimate (3) with an improved threshold on the size of the subset, namely we may remove the \u03b4 in the exponent that appears in the result from [13]. The proof of Theorem 1.5 requires estimates on the number of paths in our graph, for example Proposition 3.5. This is again done in a general way for (n, d, \u03bb)-graphs. We also note that colorful versions of Theorem 1.5 and the lemmas required to prove it can be proved with only minor modi\ufb01cations to the proof. That is, given an (n, d, \u03bb)-colored graph and a \ufb01xed coloring of a path or cycle, one can obtain the same estimates on the number of such colorful subgraphs that appear. For ease of exposition we only prove an uncolored version of Theorem 1.5, but Theorems 1.7 and 1.8 (see below) are stated and proved in a colorful way as proof of concept. It is possible through this general set up to recover Theorem 1.1 of [2] and Theorem 6 of [5]. Finally, we prove Theorem 1.5 in two di\ufb00erent ways. The second approach is quite speci\ufb01c to counting cycles, but more straightforward (it is also slightly weaker: we obtain the same quantitative results for m \u22655 but for m = 4 only prove the result up to a multiplicative constant factor). The \ufb01rst approach passes the problem to counting structures in the tensor product of two (n, d, \u03bb)-graphs. We note that the tensor product of two (n, d, \u03bb)-graphs is itself a (n2, d2, d\u03bb) graph, and so one may try to use pseudo-randomness of this graph to count subgraphs. However, this is not good enough for our purpose, and we must prove a version of the expander mixing lemma that applies speci\ufb01cally to tensor products of graphs. This result (Proposition 3.2) is signi\ufb01cantly stronger than directly applying the classical expander mixing lemma to the tensor product graph, and we believe it is of independent interest, as the second approach along with Proposition 3.2 could be used to count other structures in tensor products of pseudo-random graphs. 1.3 Distribution of disjoint trees The last question we consider in this paper is the following. Question 1.6. Let E be a set in Fd q, and T be a tree of m vertices. How large does E need to be to guarantee that the number of vertex disjoint copies of T in E is close to |E|/m? That is, we are asking for a threshold such that any set of large enough size has an almost spanning T-factor. We now to introduce the notion of the stringiness of a graph, T, denoted 6 \f\u03c3(T), which is de\ufb01ned as (d1 + 1) Qn i=2 di where d1 \u2265d2 \u00b7 \u00b7 \u00b7 \u2265dn is the degree sequence of T in nonincreasing order. Using this, Soukup [29] proved that for any tree T of m vertices with stringiness \u03c3(T), and for any E \u2282Fd q, if |E| \u226b\u03c3(T)q d+1 2 , then the number of disjoint copies of T in E is at least |U| \u03c3(T) \u2212q d+1 2 . In this section, we provide improvements of this result. Theorem 1.7. Let G be an (n, d, \u03bb)-colored graph with the color set D. Let T be a tree with edges colored by D. For any U \u2282V (G) with |U| = r \u00b7 \u03bbn d , the number of disjoint copies of H in U is at least |U| \u03c3(T) \u2212\u03bbn d , where \u03c3(T) is the stringiness of T. Theorem 1.7 directly generalizes Soukup\u2019s result in [29] to pseudo-random graphs. However, the stringiness of a tree may be exponential in the number of vertices. Using a di\ufb00erent method, we prove a theorem which for most trees does much better. Theorem 1.8. Let G be an (n, d, \u03bb)-colored graph with the color set D. Let T be a tree of m vertices with edges colored by D. For any U \u2282V (G) with |U| \u2265m(m \u22121) \u00b7 \u03bbn d , the number of disjoint copies of T in U is at least |U| m \u2212\u03bbn d . 2 Proof of Theorem 1.3 Set Vi = V (Gi) and Ei = E(Gi) for 1 \u2264i \u22642. If i satis\ufb01es \u03bbi di = max n \u03bb1 d1 , \u03bb2 d2 o then throughout the proof we will use \u03bb d to denote \u03bbi di . 2.1 Square-norm For functions f1, f2, f3, f4 : V1 \u00d7 V2 \u2192[\u22121, 1], we de\ufb01ne N(f1, f2, f3, f4) :=E a,b,c,d (a,b)\u2208E1,(c,d)\u2208E2 f1(a, c)f2(a, d)f3(b, c)f4(b, d) := 1 |V1|d1|V2|d2 X a,b,c,d (a,b)\u2208E1,(c,d)\u2208E2 f1(a, c)f2(a, d)f3(b, c)f4(b, d), 7 \fand M(f1, f2, f3, f4) :=Ea,b,c,df1(a, c)f2(a, d)f3(b, c)f4(b, d) := 1 |V1|2|V2|2 X a,b,c,d f1(a, c)f2(a, d)f3(b, c)f4(b, d). Let S be any subset of V1 \u00d7 V2. Recall that when context is clear, we use S(\u00b7) to denote the characteristic function \u03c7S on the set S. We now prove two simple but useful facts about M using Cauchy-Schwarz. Proposition 2.1. M(S, S, S, S) \u2265 \u0012 |S| |V1||V2| \u00134 . Proof. We write the de\ufb01nition of |S| as a sum and apply Cauchy-Schwarz twice to get |S| = X a\u2208V1 X b\u2208V2 S(a, b) \u2264  X a\u2208V1 12 ! 1 2 \uf8eb \uf8edX a\u2208V1  X b\u2208V2 S(a, b) !2\uf8f6 \uf8f8 1 2 = |V1| 1 2  X b\u2208V2 X c\u2208V2 X a\u2208V1 S(a, b)S(a, c) ! 1 2 \u2264|V1| 1 2 \uf8eb \uf8ec \uf8ed  X b\u2208V2 X c\u2208V2 12 ! 1 2 \uf8eb \uf8edX b\u2208V2 X c\u2208V2  X a\u2208V1 S(a, b)S(a, c) !2\uf8f6 \uf8f8 1 2\uf8f6 \uf8f7 \uf8f8 1 2 = |V1| 1 2 \uf8eb \uf8ed|V2|  X b\u2208V2 X c\u2208V2 X a\u2208V1 X d\u2208V1 S(a, b)S(a, c)S(d, b)S(d, c) ! 1 2\uf8f6 \uf8f8 1 2 , which, upon rearranging and renaming variables becomes |V1| 1 2|V2| 1 2 \u0000|V1|2|V2|2M(S, S, S, S) \u0001 1 4 . Comparing this to |S| yields the desired result. For any function f : V1 \u00d7 V2 \u2192[\u22121, 1], we de\ufb01ne ||f||\u25a1(V1\u00d7V2) := M(f, f, f, f)1/4. Lemma 2.2. For functions f1, f2, f3, f4 : V1 \u00d7 V2 \u2192[\u22121, 1], we have M(f1, f2, f3, f4) \u2264min i ||fi||\u25a1(V1\u00d7V2). 8 \fProof. We apply Cauchy-Schwarz to the de\ufb01nition of M to get M(f1, f2, f3, f4) = 1 |V1|2|V2|2 X a,b\u2208V1,c,d\u2208V2 f1(a, c)f2(a, d)f3(b, c)f4(b, d) = 1 |V1|2|V2|2 X a,b\u2208V1  X c\u2208V2 f1(a, c)f3(b, c) !  X d\u2208V2 f2(a, d)f4(b, d) ! \u2264 1 |V1|2|V2|2 \uf8eb \uf8edX a,b\u2208V1  X c\u2208V2 f1(a, c)f3(b, c) !2\uf8f6 \uf8f8 1 2 \u00b7 \uf8eb \uf8edX a,b\u2208V1  X d\u2208V2 f2(a, d)f4(b, d) !2\uf8f6 \uf8f8 1 2 = (M(f1, f1, f3, f3)) 1 2 \u00b7 (M(f2, f2, f4, f4)) 1 2 . A similar calculation using Cauchy-Schwarz and reversing the roles of V1 and V2 gives that M(f1, f2, f3, f4) \u2264(M(f1, f2, f1, f2))1/2 \u00b7 (M(f3, f4, f3, f4))1/2. We \ufb01nish by combining these inequalities and using the fact that M(fi, fi, fi, fi) \u22641 for i = 1, 2, 3, 4. 2.2 A weak hypergraph regularity lemma Let B be a \u03c3-algebra on V1 and C be a \u03c3-algebra on V2. We recall here that a \u03c3-algebra on Vi is a collection of sets in Vi that contains Vi, \u2205, and is closed under \ufb01nite intersections, unions, and complements. The complexity of a \u03c3-algebra B is the smallest number of sets (atoms) needed to generate B, and we denote by complexity(B). Notice that |B| \u22642complexity(B). We denote the smallest \u03c3-algebra on V1 \u00d7 V2 that contains both B \u00d7 V2 and V1 \u00d7 C by B \u2228C. For a function f : V1 \u00d7 V2 \u2192R, we de\ufb01ne the conditional expectation E(f|B \u2228C): V \u2192R by the formula E(f|B \u2228C)(x) := 1 |(B \u2228C)(x)| X y\u2208(B\u2228C)(x) f(y), where (B \u2228C)(x) denotes the smallest element of B \u2228C that contains x. We note that an atom of B \u2228C has the form U \u00d7 V where U and V are atoms of B and C, respectively. The following lemma is a special case of Lemma 2.2 in [25]. We refer the reader to [25] for a detailed proof. Lemma 2.3. For any \u03f5 > 0, there exist \u03c3-algebras B on V1 and C on V2 such that each 9 \falgebra is spanned by at most O(\u03f5\u22128) sets, and ||S \u2212E(S|B \u2228C)||\u25a1(V1\u00d7V2) \u2264\u03f5. We recall that E(S|B \u2228C)(x) = |S \u2229(B \u00d7 C)| |B||C| , where B \u00d7 C is the atom of B \u2228C containing x. 2.3 A generalized von-Neumann type estimate Lemma 2.4. For functions f1, f2, f3, f4 : V1 \u00d7 V2 \u2192[\u22121, 1], we have |N(f1, f2, f3, f4)| \u2264min j ||fj||\u25a1(V1\u00d7V2) + O \u0012\u03bb1/4 d1/4 \u0013 . To prove this lemma, we recall the following expander mixing lemma. Lemma 2.5. Let G = (V, E) be an (n, d, \u03bb)-graph, and A be its adjacency matrix. For real f, g \u2208L2(V ), we have |\u27e8f, Ag\u27e9\u2212d|V |E(f)E(g)| \u2264\u03bb\u2225f\u22252\u2225g\u22252, where E(f) := 1 |V | X v\u2208V f(v), ||f||2 2 = X v\u2208V |f(v)|2. Proof of Lemma 2.4. Set \u03c3i(x, y) = |Vi|/di if (x, y) \u2208Ei and 0 otherwise and let Ex,y\u2208Vi := 1 |Vi|2 P x,y\u2208Vi. Then for f, g: Vi \u2192[\u22121, 1], by using the expander mixing lemma, one has X x\u223cy f(x)g(y) = \u27e8f, Ag\u27e9\u2264di |Vi| X x,y f(x)g(y) + \u03bbi\u2225f\u22252\u2225g\u22252. Dividing both sides by di|Vi| and using \u2225f\u22252\u2225g\u22252 \u2264|Vi| gives Ex,y\u2208Vif(x)g(y)\u03c3i(x, y) =   1 |Vi|2 X x,y f(x)g(y) ! + \u03bbi di = Ex,yf(x)f(y) + \u03bbi di . Thus, |Ex,yf(x)g(y)\u03c3i(x, y)|2 \u2264(Ex,y,z,tf(x)g(z)f(y)g(t)) + 2\u03bbi di Ex,yf(x)g(y) + \u03bb2 i d2 i \u2264Ex,yf(x)f(y) + 3\u03bbi di , 10 \fwhere we have used the fact that Ez,tg(z)g(t), Ex,yf(x)g(y) \u22641. In other words, for functions f, g : Vi \u2192[\u22121, 1], we have |Ex,yf(x)g(y)\u03c3i(x, y)|2 \u2264Ex,yf(x)f(y) + 3\u03bbi di . (4) The same holds when we switch between f and g: |Ex,yf(x)g(y)\u03c3i(x, y)|2 \u2264Ex,yg(x)g(y) + 3\u03bbi di . (5) In the next step, we want to show that N(f1, f2, f3, f4) \u2264||f1||\u25a1(V1\u00d7V2) + O \u0012\u03bb d \u0013 . Using the de\ufb01nitions, we have N(f1, f2, f3, f4) = Ea,b,c,df1(a, c)f2(a, d)f3(b, c)f4(b, d)\u03c31(a, b)\u03c32(c, d). For a \ufb01xed pair (c, d), set fc,d(a) = f1(a, c)f2(a, d) and gc,d(b) = f3(b, c)f4(b, d). Then we have |N(f1, f2, f3, f4)|2 =   1 |V1|2|V2|2 X a,b,c,d fc,d(a)gc,d(b)\u03c31(a, b)\u03c32(c, d) !2 \u2264   1 |V1|2|V2|2 X c,d p \u03c32(c, d) \f \f \f \f \f p \u03c32(c, d) X a,b fc,d(a)gc,d(b)\u03c31(a, b) \f \f \f \f \f !2 \u2264 1 |V1|4|V2|4  X c,d \u03c32(c, d) ! \uf8eb \uf8edX c,d \u03c32(c, d) \f \f \f \f \f X a,b fc,d(a)gc,d(b)\u03c31(a, b) \f \f \f \f \f 2\uf8f6 \uf8f8 = (Ec,d\u03c32(c, d)) \u0000Ec,d\u03c32(c, d)|Ea,bfc,d(a)gc,d(b)\u03c31(a, b)|2\u0001 = Ec,d\u03c3(c, d)|Ea,bfc,d(a)gc,d(b)\u03c3(a, b)|2, where the \ufb01rst inequality uses the triangle inequality and rearranging, the second inequality is Cauchy-Schwarz, the next line is rearranging, and the last equality uses the fact that Ec,d\u03c32(c, d) = 1 |V2|2 \u00b7 |V2| d2 \u00b7 |V2| \u00b7 d2 = 1. Therefore, the inequality (4) implies |N(f1, f2, f3, f4)|2 \u2264Ec,d\u03c32(c, d) \u0012 Ea,bfc,d(a)fc,d(b) + 3\u03bb1 d1 \u0013 (6) = Ea,b,c,d\u03c32(c, d)fc,d(a)fc,d(b) + 3 (Ec,d\u03c32(c, d)) \u03bb1 d1 = (Ea,b,c,df1(a, c)f1(b, c)f2(a, d)f2(b, d)\u03c32(c, d)) + 3\u03bb1 d1 . 11 \fBy another similar argument with \u02c6 fa,b(c) = f1(a, c)f1(b, c) and \u02c6 ga,b(d) = f2(a, d)f2(b, d) for each \ufb01xed pair (a, b), we have |N(f1, f2, f3, f4)|4 \u2264 \u0012 Ea,b,c,d \u02c6 fa,b(c)\u02c6 ga,b(d)\u03c32(c, d) + 3\u03bb1 d1 \u00132 \u2264 \u0010 Ea,b,c,d \u02c6 fa,b(c)\u02c6 ga,b(d)\u03c32(c, d) \u00112 + 15\u03bb1 d1 , using that Ea,b,c,d \u02c6 fa,b(c)\u02c6 ga,b(d)\u03c32(c, d) \u22641 and \u03bb1/d1 \u22641. Now \u0010 Ea,b,c,d \u02c6 fa,b(c)\u02c6 ga,b(d)\u03c32(c, d) \u00112 = 1 |V1|4|V2|4  X a,b 1 X c,d \u02c6 fa,b(c)\u02c6 ga,b(d)\u03c32(c, d) !2 \u2264 1 |V1|2 X a,b   1 |V2|2 X c,d \u02c6 fa,b(c)\u02c6 ga,b(d)\u03c32(c, d) !2 \u2264 1 |V1|2 X a,b \u0012 Ec,d \u02c6 fa,b(c) \u02c6 fa,b(d) + 3\u03bb2 d2 \u0013 =Ea,b \u0012 Ec,df1(a, c)f1(b, c)f1(a, d)f1(b, d) + 3\u03bb2 d2 \u0013 , by Cauchy-Schwarz and (4) respectively. As a consequence, we obtain |N(f1, f2, f3, f4)|4 \u2264Ea,b,c,df1(a, c)f1(b, c)f1(a, d)f1(b, d) + 15\u03bb1 d1 + 3\u03bb2 d2 = M(f1, f1, f1, f1) + O \u0012\u03bb1 d1 + \u03bb2 d2 \u0013 . Notice that the same holds when f1 on the right hand side is replaced by fi for 2 \u2264i \u22644. In short, |N(f1, f2, f3, f4)| \u2264min j ||fj||\u25a1(V1\u00d7V2) + O \u0012\u03bb1/4 d1/4 \u0013 . This completes the proof. With Lemmas 2.3 and 2.4 in hand, we are ready to prove Theorem 1.3. Proof of Theorem 1.3: For any \u03f5 > 0, by Lemma 2.3, we can see that there exist \u03c3algebras B and C on V1 and V2, respectively, with complexity bounded above by O (\u03f5\u22128) , so that ||S \u2212E(S|B \u2228C)||\u25a1(V1\u00d7V2) \u2264\u03f5. (7) Let g denote E(S|B \u2228C), and de\ufb01ne h(x) := S(x) \u2212g(x). 12 \fTherefore, (7) gives us that ||h||\u25a1(V1\u00d7V2) \u2264\u03f5. (8) Both g and h are functions from V1 \u00d7 V2 to the interval [\u22121, 1]. Recalling the de\ufb01nition of N above, we see that N(S, S, S, S) = N(g, g, g, g) + N(h, h, h, h) + R, where R is a sum over all expressions of the form N(f1, f2, f3, f4), where the fj in each term are either g or h, but not all the same. Speci\ufb01cally, set \u2126:= {g, h}4 \\{(g, g, g, g), (h, h, h, h)}, denote a quadruple of functions by F = (f1, f2, f3, f4) \u2208\u2126, and write R = X F\u2208\u2126 Ea,b,c,df1(a, c)f2(a, d)f3(b, c)f4(b, d)E1(a, b)E2(c, d), where Ei(x, y) is the indicator that xy \u2208E(Gi). Combining Lemma 2.4 and (8) gives |N(h, h, h, h)| \u2264||h||\u25a1(V1\u00d7V2) + O \u0012\u03bb1/4 d1/4 \u0013 \u2264\u03f5 + O \u0012\u03bb1/4 d1/4 \u0013 . Similarly, for any other choice of F \u2208\u2126, we must have h in at least one entry, so we get |N(F)| \u2264min j ||fj||\u25a1(V1\u00d7V2) + O \u0012\u03bb1/4 d1/4 \u0013 \u2264||h||\u25a1V1\u00d7V2 + O \u0012\u03bb1/4 d1/4 \u0013 \u2264\u03f5 + O \u0012\u03bb1/4 d1/4 \u0013 . Putting these together we get that |N(S, S, S, S) \u2212N(g, g, g, g)| = O \u0012 \u03f5 + \u03bb1/4 d1/4 \u0013 (9) Similarly, by Lemma 2.2, we know that M(F) \u2264min i ||fi||\u25a1(V1\u00d7V2), so we get that |M(S, S, S, S) \u2212M(g, g, g, g)| = O (\u03f5) . (10) By de\ufb01nition, g is a linear combination of indicator functions of atoms of the \u03c3-algebra B\u2228C. By Lemma 2.3, we know that there is some positive constant c > 0 so that the number of terms in this linear combination is no more than 2c\u03f5\u22128. So we can write N(g, g, g, g) as a 13 \flinear combination of terms of the form N(B1 \u00d7 C1, B2 \u00d7 C2, B3 \u00d7 C3, B4 \u00d7 C4) = Ea,b,c,d(B1 \u00d7 C1)(a, c) \u00b7 (B2 \u00d7 C2)(a, d) \u00b7 (B3 \u00d7 C3)(b, c) \u00b7 (B4 \u00d7 C4)(b, d)\u03c31(a, b)\u03c32(c, d), for some atoms Bj\u00d7Cj (and their indicator functions) in B\u00d7C. Here as before we use \u03c3i(x, y) equals |Vi|/di if {x, y} \u2208Ei and 0 otherwise. However, if we split this up by variables, we get that N(B1 \u00d7 C1, B2 \u00d7 C2, B3 \u00d7 C3, B4 \u00d7 C4) = Ea,b,c,d(B1 \u2229B2)(a) \u00b7 (B3 \u2229B4)(b) \u00b7 (C1 \u2229C3)(c) \u00b7 (C2 \u2229C4)(d)\u03c31(a, b)\u03c32(c, d) = (Ea,b(B1 \u2229B2)(a) \u00b7 (B3 \u2229B4)(b)\u03c31(a, b)) (Ec,d(C1 \u2229C3)(c) \u00b7 (C2 \u2229C4)(d)\u03c32(c, d)) . By applying the expander mixing lemma as in the proof of Lemma 2.4, we see N(B1 \u00d7 C1, B2 \u00d7 C2, B3 \u00d7 C3, B4 \u00d7 C4) = \u0012 Ea,b(B1 \u2229B2)(a) \u00b7 (B3 \u2229B4)(b) + O \u0012\u03bb1 d1 \u0013\u0013 \u0012 Ec,d(C1 \u2229C3)(c) \u00b7 (C2 \u2229C4)(d) + O \u0012\u03bb2 d2 \u0013\u0013 = M(B1 \u00d7 C1, B2 \u00d7 C2, B3 \u00d7 C3, B4 \u00d7 C4) + O \u0012\u03bb d \u0013 , where the last line uses the de\ufb01nition of M and that each expectation is at most 1. Since g is a linear combination of at most 2c\u03f5\u22128 terms, we see that |N(g, g, g, g) \u2212M(g, g, g, g)| = O \u0012 2c\u2032\u03f5\u22128 \u03bb d \u0013 , for some positive constant c\u2032. Using (9) followed by the previous estimate and (10), we get that for some constant k > 0, we have N(S, S, S, S) \u2265N(g, g, g, g) \u2212k\u03f5 \u2212k\u03bb1/4 d1/4 \u2265M(g, g, g, g) \u2212k2c\u2032\u03f5\u22128 \u03bb d \u2212k\u03f5 \u2212k\u03bb1/4 d1/4 \u2265M(S, S, S, S) \u2212k\u03f5 \u2212k2c\u03f5\u22128 \u03bb d \u2212k\u03f5 \u2212k\u03bb1/4 d1/4 . Now applying Proposition 2.1 to this estimate gives us N(S, S, S, S) \u2265 \u0012 |S| |V1||V2| \u00134 \u22122k\u03f5 \u2212k2c\u03f5\u22128 \u03bb d \u2212k\u03bb1/4 d1/4 . (11) Recall that by assumption, |S| \u2265\u03b4|V1||V2|, so to guarantee that N = N(S, S, S, S) is positive, we just need to pick \u03f5 so that the right-hand-side of (11) is bigger than \u03b4\u20324, or 14 \fequivalently, \u03b44 \u2212\u03b4\u20324 \u22652k\u03f5 + k2c\u03f5\u22128 \u03bb d + k\u03bb1/4 d1/4 . 3 Proof of Theorem 1.5 To prove Theorem 1.5, we present two approaches based on two counting lemmas. While the second counting lemma is a direct consequence of the expander mixing lemma for a single graph, the \ufb01rst counting lemma is a stronger and more practical variant for tensor of two pseudo-random graphs, which is quite interesting on its own. 3.1 The \ufb01rst counting lemma for cycles Let us brie\ufb02y describe the ideas of counting cycles here. Assume we want to count the number of cycles of length 2k for some integer k \u22652. Given four vertices x, y, z, w, if x and y are connected by a path of length k \u22121, and the same happens for z and w, then we will have a cycle of length 2k of the form x \u2212yw \u2212zx (Figure 1) when there are edges between x and z, and between y and w. Thus, the problem is reduced to counting the number of pairs of edges between the endpoints of pairs of paths of length k \u22121. Figure 1: Counting pairs of edges xz and yw. To this end, we make use of the notation of tensor of two pseudo-random graphs. For two graphs G1 = (V1, E1) and G2 = (V2, E2), the tensor product G1 \u2297G2 is a graph with vertex set V (G1 \u2297G2) = V1 \u00d7 V2, and there is an edge between (u, v) and (u\u2032, v\u2032) if and only if (u, u\u2032) \u2208E1 and (v, v\u2032) \u2208E2. Suppose that the adjacency matrices of G1 and G2 are A and B, respectively, then the adjacency matrix of G1 \u2297G2 is the tensor product of A and B. It is well-known that if \u03b31, . . . , \u03b3n are eigenvalues of A and \u03b3\u2032 1, . . . , \u03b3\u2032 m are eigenvalues of B, then the eigenvalues of A \u2297B are \u03b3i\u03b3\u2032 j with 1 \u2264i \u2264n, 1 \u2264j \u2264m (see [26] for more details). It is not hard to use the expander mixing lemma to get the following. 15 \fProposition 3.1. Let G be an (n, d, \u03bb)-graph. For two non-negative functions f, g: V \u00d7V \u2192 R, we have \f \f \f \f \f \f X (x,z)\u2208E,(y,w)\u2208E f(x, y)g(z, w) \u2212d2 n2||f||1||g||1 \f \f \f \f \f \f \u2264d\u03bb||f||2||g||2. Our \ufb01rst counting lemma o\ufb00ers better bounds as follows. Proposition 3.2. (First counting lemma) Let G be an (n, d, \u03bb)-graph. For two non-negative functions f, g: V \u00d7 V \u2192R, we de\ufb01ne F(x) = P y f(x, y), G(z) = P w g(z, w), F \u2032(y) = P x f(x, y), and G\u2032(w) = P z g(z, w). Then we have \f \f \f \f \f \f X (x,z)\u2208E,(y,w)\u2208E f(x, y)g(z, w) \u2212d2 n2||f||1||g||1 \f \f \f \f \f \f \u2264\u03bb2||f||2||g||2+d\u03bb n2 (||F||2||G||2 + ||F \u2032||2||G\u2032||2) . Proof. Suppose G is a d-regular graph on vertex set V with |V | = n, and let A denote its adjacency matrix. For two real-valued functions f, g: V \u00d7 V \u2192R, we de\ufb01ne \u27e8f, g\u27e9= X (v1,v2)\u2208V \u00d7V f(v1, v2)g(v1, v2), and ||f||2 2 = \u27e8f, f\u27e9. We denote the set of all real-valued functions on V \u00d7 V by L2(V \u00d7 V ). For the remainder of the proof we will assume that f, g \u2208L2(V \u00d7 V ) are non-negative functions. We de\ufb01ne A \u2297Af(v1, v2) = X (u1,u2): (u1,v1)\u2208E,(u2,v2)\u2208E f(u1, u2). That is, A \u2297A is the adjacency matrix of G \u2297G. In the remainder, we denote A \u2297A by B. Let \u03bb1 \u2265\u03bb2 \u2265\u00b7 \u00b7 \u00b7 \u2265\u03bbn be the eigenvalues of A corresponding to eigenfunctions e1, \u00b7 \u00b7 \u00b7 , en. Without loss of generality, assume that the ei form an orthonormal basis of Rn. Then the eigenfunctions of B are exactly ei \u2297ej for all 1 \u2264i, j \u2264n corresponding to eigenvalue \u03bbij := \u03bbi\u03bbj. We observe that f = X i,j \u27e8f, ei \u2297ej\u27e9ei \u2297ej. So Bg = X i,j \u27e8Bg, ei \u2297ej\u27e9ei \u2297ej = X ei\u2297ej \u03bbij\u27e8g, ei \u2297ej\u27e9ei \u2297ej 16 \fWe note that A has a constant eigenfunction that will be denoted by e1, i.e. e1(v) = 1/\u221an, \u2200v \u2208V. This means that B also has constant eigenfunction de\ufb01ned by e1 \u2297e1(u, v) = 1/n \u2200(u, v) \u2208V \u00d7 V. We have X (x,z)\u2208E,(y,w)\u2208E f(x, y)g(z, w) = \u27e8f, Bg\u27e9= X i,j \u03bbij\u27e8g, ei \u2297ej\u27e9\u27e8f, ei \u2297ej\u27e9. De\ufb01ne S1 :=\u03bb11\u27e8g, e1 \u2297e1\u27e9\u27e8f, e1 \u2297e1\u27e9 S2 := n X j=2 \u03bb1j\u27e8g, e1 \u2297ej\u27e9\u27e8f, e1 \u2297ej\u27e9 S3 := n X i=2 \u03bbi1\u27e8g, ei \u2297e1\u27e9\u27e8f, ei \u2297e1\u27e9 S4 := n X i,j=2 \u03bbij\u27e8g, ei \u2297ej\u27e9\u27e8f, ei \u2297ej\u27e9. And so X (x,z)\u2208E,(y,w)\u2208E f(x, y)g(z, w) \u2212S1 = S2 + S3 + S4. We now estimate each Si. Since \u03bb1 = d and e1 is constant, it is easy to see that S1 = \u03bb11 \u001c f, 1 n1 \u001d \u001c g, 1 n1 \u001d = d2 n2||f||1||g||1. For S4, if i, j > 1 we have that \u03bbij \u2264\u03bb2 and hence S4 \u2264\u03bb2 n X i,j=2 \u27e8g, ei \u2297ej\u27e9\u27e8f, ei \u2297ej\u27e9\u2264\u03bb2   n X i,j=2 \u27e8g, ei \u2297ej\u27e92 !1/2   n X i,j=2 \u27e8f, ei \u2297ej\u27e92 !1/2 \u2264\u03bb2   n X i,j=1 \u27e8g, ei \u2297ej\u27e92 !1/2   n X i,j=1 \u27e8f, ei \u2297ej\u27e92 !1/2 = \u03bb2||f||2||g||2, where the second inequality follows by Cauchy-Schwarz. 17 \fTo estimate S2, note that \u03bb1j \u2264\u03bbd, and e1 \u2297ej(v1, v2) = 1 \u221anej(v2). Using Cauchy-Schwarz, we have that S2 \u2264\u03bbd n X j=2 \u27e8g, e1 \u2297ej\u27e9\u27e8f, e1 \u2297ej\u27e9\u2264\u03bbd  n X j=2 \u27e8g, e1 \u2297ej\u27e92 !1/2  n X j=2 \u27e8f, e1 \u2297ej\u27e92 !1/2 \u2264\u03bbd  n X j=1 \u27e8g, e1 \u2297ej\u27e92 !1/2  n X j=1 \u27e8f, e1 \u2297ej\u27e92 !1/2 To estimate this quantity, note that \u27e8g, e1 \u2297ej\u27e9= X u,v g(u, v)e1 \u2297ej(u, v) = 1 \u221an X u,v g(u, v)ej(v) = 1 \u221an X v G\u2032(v)ej(v), and similarly \u27e8f, e1 \u2297ej\u27e9= 1 \u221an P v F \u2032(v)ej(v). Therefore, we have that n X j=1 \u27e8g, e1 \u2297ej\u27e92 = n X j=1 1 n X u,v G\u2032(u)G\u2032(v)ej(u)ej(v) = 1 n X u,v   G\u2032(u)G\u2032(v) n X j=1 ej(u)ej(v) ! . Now notice that because the ei form an orthonormal basis, we have that n X j=1 ej(u)ej(v) = \uf8f1 \uf8f2 \uf8f3 1 u = v 0 u \u0338= v. Hence we have n X j=1 \u27e8g, e1 \u2297ej\u27e92 = 1 n n X u=1   (G\u2032(u))2 n X j=1 ej(u)2 ! = 1 n n X u=1 (G\u2032(u))2 = 1 n||G\u2032||2 2. Similarly Pn j=1\u27e8f, e1 \u2297ej\u27e92 = 1 n||F \u2032||2 2. Combining everything we have that S2 \u2264\u03bbd n2 ||G\u2032||2||F \u2032||2. A symmetric proof shows that S3 \u2264\u03bbd n2 ||G||2||F||2. 18 \f3.2 The second counting lemma for cycles Assume we want to count the number of cycles of length 2k for some integer k \u22651. Our second strategy for cycles can be explained as follows. Given three vertices x, y, and z, if x and y are connected by a path of length k, and x and z are connected by a path of length k \u22121, then we have a cycle of length 2k of the form x \u2212yz \u2212x if and only if y and z are adjacent (Figure 2). So the problem is reduced to counting the number of edges between the endpoints of pairs of paths pinned at a vertex. Figure 2: Counting edges yz. Proposition 3.3. (Second counting lemma) Let G be an (n, d, \u03bb)-graph. Let U be a set of vertices in G. For any two vertices x and y, let pk(x, y) be the number of paths of length k between x and y with vertices in between belonging to U. Then we have \f \f \f \f \f \f C2k+1(U) \u2212d n X x\u2208U  X y\u2208U pk(x, y) !2\f \f \f \f \f \f \u2264\u03bb X x,y\u2208U pk(x, y)2, and \f \f \f \f \fC2k(U) \u2212d n X x\u2208U  X y\u2208U pk(x, y) ! \u00b7  X z\u2208U pk\u22121(x, z) !\f \f \f \f \f \u2264\u03bb X x\u2208U  X y\u2208U pk(x, y)2 !1/2 \u00b7  X z\u2208U pk\u22121(x, z)2 !1/2 . Proof. We \ufb01rst observe that the number of odd cycles of length 2k + 1 in U is equal to the sum X x,y,z\u2208U3,(y,z)\u2208E(G) pk(x, y)pk(x, z). Given x \u2208U, set f(y) = U(y)pk(x, y), then the above sum can be rewritten as X x\u2208U X (y,z)\u2208E(G) f(y)f(z). 19 \fApplying Lemma 2.5, the \ufb01rst statement is proved. For the second statement, as above, the number of even cycles of length 2k in U is equal to the sum X x,y,z\u2208U3,(y,z)\u2208E(G) pk(x, y)pk\u22121(x, z). Given x \u2208U, set f(y) = U(y)pk(x, y) and g(z) = U(z)pk\u22121(x, z), then the above sum can be rewritten as X x\u2208U X (y,z)\u2208E(G) f(y)g(z). Applying Lemma 2.5, the proposition is proved. Using the facts that X x\u2208U  X y\u2208U pk(x, y) !2 = P2k(U), X x,y\u2208U pk(x, y)2 = C2k(U), X x\u2208U  X y\u2208U pk(x, y) ! \u00b7  X z\u2208U pk\u22121(x, z) ! = P2k\u22121(U), and the following application of Cauchy-Schwarz, X x\u2208U  X y\u2208U pk(x, y)2 !1/2 \u00b7  X z\u2208U pk\u22121(x, z)2 !1/2 \u2264(C2k(U)C2k\u22122(U))1/2 , one derives the following corollary. Corollary 3.4. Let G be an (n, d, \u03bb)-graph. Let U be a set of vertices in G. Then \f \f \f \fC2k+1(U) \u2212d nP2k(U) \f \f \f \f \u2264\u03bbC2k(U), and \f \f \f \fC2k(U) \u2212d nP2k\u22121(U) \f \f \f \f \u2264\u03bb (C2k(U)C2k\u22122(U))1/2 . 3.3 Distribution of paths We have seen that to apply the two counting lemmas, we need to have estimates on the paths of a given length in a vertex set. We now provide relevant results on paths. Proposition 3.5. Let G be an (n, d, \u03bb)-graph, k \u22651 an integer, and U be a vertex set with 20 \f\u03bb \u00b7 n d = o(|U|). Let Pk(U) denotes the number of paths of length k in U. Then we have Pk(U) = \u0014 1 + \u0398 \u0012 \u03bbn d|U| \u0013\u0015 |U|k+1dk nk . Proof. We \ufb01rst prove the following two estimates: \f \f \f \fP2k+1(U) \u2212dPk(U)2 n \f \f \f \f \u2264\u03bbP2k(U), (12) and \f \f \f \fP2k(U) \u2212dPk(U)Pk\u22121(U) n \f \f \f \f \u2264\u03bb p P2k(U)P2k\u22122(U). (13) For u \u2208U, let f(u) be the number of paths of length k of the form (u1, . . . , uk, u) where ui \u2208U. Similarly, for v \u2208U, let g(v) be the number of paths of length k of the form (v1, . . . , vk, v) where vi \u2208U. To use Lemma 2.5 we need to estimate the norms and the inner product. We have that the adjacency matrix A acts on g by the formula Ag(u) = X (u,v)\u2208E(G) g(v) which is the number of paths of length k + 1 of the form (v1, . . . , vk, v, u). For the inner product, we have \u27e8f, Ag\u27e9= X u\u2208V (G) f(u)Ag(u) = X u\u2208V (G) f(u)Ag(u) = P2k+1(U). It is clear that E(f) = E(g) = 1 |V | \u00b7 Pk(U). and ||f||2 2 = ||g||2 2 = P2k(U). Applying Lemma 2.5 we have that \f \f \f \f \fP2k+1(U) \u2212d|V | \u0012 1 |V | \u00b7 Pk(U) \u00132\f \f \f \f \f \u2264\u03bbP2k(U) which is equivalent to (12). The estimate (13) also follows from a similar argument with the same f and g(v) de\ufb01ned to be the number of paths of length k\u22121 of the form (v1, . . . , vk\u22121, v). We now proceed by induction on k. The case k = 0 is trivial and the case k = 1 follows from Lemma 2.5 and the estimate (13). Suppose that the statement holds for all 2k \u22651. We now show that it also holds for 2k + 1 21 \fand 2k +2. Indeed, it follows from the estimate (12) and induction hypothesis that we have P2k+1(U) \u2264d nPk(U)2 + \u03bbP2k(U) \u2264d n |U|2k+2d2k n2k \u0012 1 + O \u0012 \u03bbn d|U| \u0013\u00132 + \u03bb|U|2k+1d2k n2k \u0012 1 + O \u0012 \u03bbn d|U| \u0013\u0013 = |U|2k+2d2k+1 n2k+1 \u0012 1 + O \u0012 \u03bbn d|U| \u0013\u0013 whenever |U| \u226b\u03bb n d. The lower bound follows in the same way. For the case 2k + 2, it also follows from the estimate (13) that P2k+2(U) \u2264dPk(U)Pk+1(U) n + \u03bb p P2k(U)P2k+2(U). Solving this inequality in x = p P2k+2(U), we obtain P2k+2(U) \u2264 \uf8eb \uf8ed\u2212\u03bb p P2k(U) + q \u03bb2P2k(U) + 4dPk(U)Pk+1(U) n 2 \uf8f6 \uf8f8 2 . Using the induction hypothesis and that \u03bbn d = o(|U|), we have that \u03bb2P2k(U) = o \u0012dPk(U)Pk+1(U) n \u00b7 \u03bbn d|U| \u0013 and that \u03bb p P2k(U) r dPk(U)Pk+1(U) n = O \u0012|U|2k+3d2k+2 n2k+2 \u00b7 \u03bbn d|U| \u0013 . Hence the entire expression is bounded above by |U|2k+3d2k+2 n2k+2 \u0014 1 + O \u0012 \u03bbn d|U| \u0013\u0015 . Using lower bounds of the estimates (12) and (13), and an identical argument also gives us Pk(U) \u2265 \u0014 1 \u2212O \u0012 \u03bbn d|U| \u0013\u0015 |U|k+1 \u0012d n \u0013k , under the condition \u03bb n d = o(|U|). This completes the proof of the proposition. 3.4 Proof of Theorem 1.5 using the \ufb01rst counting lemma Proof of Theorem 1.5. We proceed by induction on m. We \ufb01rst start with the base case m = 4. 22 \fLet f, g : U \u00d7 U \u2192R de\ufb01ned by f(x, y) = \uf8f1 \uf8f2 \uf8f3 1, if (x, y) \u2208E(G) 0, otherwise and g(z, w) = \uf8f1 \uf8f2 \uf8f3 1, if (z, w) \u2208E(G) 0, otherwise. It is clear that C4(U) = P (x,z),(y,w)\u2208E(G) f(x, y)g(z, w). To apply Proposition 3.2, we need to check the norms of functions f, g, F, G, F \u2032, and G\u2032. Using Proposition 3.5, we have ||f||1 = X x,y\u2208U f(x, y) = P1(U) = \u0012 1 + \u0398 \u0012 \u03bbn d|U| \u0013\u0013 |U|2 d n, ||f||2 =  X x,y\u2208U f(x, y)2 !1/2 =  X x,y\u2208U f(x, y) !1/2 = (P1(U))1/2 = \u0012 1 + \u0398 \u0012 \u03bbn d|U| \u0013\u0013 |U| r d n, using the Taylor series for \u221a1 + x and the assumption that \u03bbn d = o(|U|). Similarly, ||g||1 = \u0012 1 + \u0398 \u0012 \u03bbn d|U| \u0013\u0013 |U|2 d n, and ||g||2 = \u0012 1 + \u0398 \u0012 \u03bbn d|U| \u0013\u0013 |U| r d n. For functions F, G, F \u2032, G\u2032 de\ufb01ned as in Proposition 3.2, we have that ||F||2 =  X x\u2208U F(x)2 !1/2 = (P2(U))1/2 = \u0012|U|3d2 n2 \u0012 1 + \u0398 \u0012 \u03bbn d|U| \u0013\u0013\u00131/2 . Similarly, ||G||2 = ||F \u2032||2 = ||G\u2032||2 = \u0012|U|3d2 n2 \u0012 1 + \u0398 \u0012 \u03bbn d|U| \u0013\u0013\u00131/2 . Substituting these estimates into Proposition 3.2, we have that \f \f \f \fC4(U) \u2212 \u0012 1 + \u0398 \u0012 \u03bbn d|U| \u0013\u0013 |U|4 d4 n4 \f \f \f \f \u2264\u03bb2|U|2d n \u0012 1 + O \u0012 \u03bbn d|U| \u0013\u0013 +2\u03bb|U|3d3 n4 \u0012 1 + O \u0012 \u03bbn d|U| \u0013\u0013 . Using the assumption that \u03bbn d = o(|U|) completes this case. Assume that the statement holds for any cycle of length smaller than m \u22121, we now show that it holds for cycles of length m. We fall into two cases: Case 1: m = 2k + 1. 23 \fAs above, for x, y \u2208U, we de\ufb01ne f(x, y) = the number of paths of length k between x and y, and g(x, y) = the number of paths of length k \u22121 between x and y. Then ||f||1 = Pk(U) = \u0012 1 + \u0398 \u0012 \u03bbn d|U| \u0013\u0013 dk nk |U|k+1, ||f||2 2 = C2k(U) = (1 + o(1))|U|2kd2k n2k + O \u0012\u03bb2k\u22122|U|2d n \u0013 , ||F||2 2 = ||F \u2032||2 2 = P2k(U) = (1 + o(1))d2k n2k |U|2k+1, and ||g||1 = Pk\u22121(U) = \u0012 1 + \u0398 \u0012 \u03bbn d|U| \u0013\u0013 dk\u22121 nk\u22121|U|k, ||g||2 2 = C2(U) = (1 + o(1))|U|2d n if m = 5 ||g||2 2 = C2k\u22122(U) = (1 + o(1))|U|2k\u22122d2k\u22122 n2k\u22122 + O \u0012\u03bb2k\u22124|U|2d n \u0013 if m \u22657 ||G||2 2 = ||G\u2032||2 2 = P2k\u22122(U) = (1 + o(1))d2k\u22122 n2k\u22122|U|2k\u22121 Applying Proposition 3.2, \f \f \f \fC2k+1(U) \u2212d2 n2Pk(U)Pk\u22121(U) \f \f \f \f \u2264\u03bb2p C2k(U)C2k\u22122(U) + 2d\u03bb n2 ( p P2k(U)P2k\u22122(U)) When m = 5, we have \f \f \f \fC5(U) \u2212 \u0012 1 + \u0398 \u0012 \u03bbn d|U| \u0013\u0013 |U|5d5 n5 \f \f \f \f = O   \u03bb2 r |U|6d5 n5 + \u03bb2|U|4d2 n2 + \u03bbd4|U|4 n5 ! . Note that \u03bbd4|U|4 n5 = o \u0010 \u03bbn d|U| |U|5d5 n5 \u0011 . If |U| \u2264\u03bbn3/2 d3/2 , then the second term in the square root is bigger than the \ufb01rst, and hence \u03bb2 r |U|6d5 n5 + \u03bb2|U|4d2 n2 = O \u0012\u03bb3|U|2d n \u0013 . 24 \fIf |U| \u2265\u03bbn3/2 d3/2 then the \ufb01rst term is bigger than the second and we have \u03bb2 r |U|6d5 n5 + \u03bb2n4d2 n2 = O \u0012\u03bb2|U|3d5/2 n5/2 \u0013 . Using the assumption that |U| \u2265\u03bbn3/2 d3/2 gives that \u03bb2|U|3d5/2 n5/2 \u2264\u03bb|U|4d4 n4 = \u03bbn d|U| |U|5d5 n5 . In either case the inequality is satis\ufb01ed. For m \u22657 we have \f \f \f \fC2k+1(U) \u2212 \u0012 1 + \u0398 \u0012 \u03bbn d|U| \u0013\u0013 |U|2k+1d2k+1 n2k+1 \f \f \f \f = O   \u03bb2 s\u0014|U|2kd2k n2k + \u03bb2k\u22122|U|2d n \u0015 \u0014|U|2k\u22122d2k\u22122 n2k\u22122 + \u03bb2k\u22124|U|2d n \u0015 + \u03bb|U|2kd2k n2k+1 ! . First note that \u03bb|U|2kd2k n2k+1 = o \u0010 \u03bbn d |U|2k+1d2k+1 n2k+1 \u0011 so we may ignore this term. Hence if each of the four terms |U|4k\u22122d4k\u22122 n4k\u22122 , \u03bb2k\u22122|U|2kd2k\u22121 n2k\u22121 , \u03bb2k\u22124|U|2k+2d2k+1 n2k+1 , \u03bb4k\u22126|U|4d2 n2 , is either O \u0012|U|4kd4k \u03bb2n4k \u0013 or O \u0012\u03bb4k\u22126|U|4d2 n2 \u0013 , then we are done. The fourth term trivially satis\ufb01es the inequality. The \ufb01rst term is o \u0010 |U|4kd4k \u03bb2n4k \u0011 and \u03bb2k\u22122|U|2kd2k\u22121 n2k\u22121 = o \u0012\u03bb2k\u22124|U|2k+2d2k+1 n2k+1 \u0013 by the assumption that \u03bbn d = o(|U|). Finally, if |U| \u2265\u03bb \u0000 n d \u0001(2k\u22121)/(2k\u22122) then \u03bb2k\u22124|U|2k+2d2k+1 n2k+1 \u2264|U|4kd4k \u03bb2n4k . Otherwise \u03bb2k\u22124|U|2k+2d2k+1 n2k+1 \u2264\u03bb4k\u22126|U|4d2 n2 . Case 2: m = 2k. For this case, we want to apply Proposition 3.2 again, so we need to de\ufb01ne suitable functions f and g, namely, for x, y \u2208U, f(x, y) = g(x, y) = the number of paths of length k \u22121 between x and y. 25 \fThen, by inductive hypothesis and Proposition 3.5, one has ||f||1 = ||g||1 = Pk\u22121(U) = \u0012 1 + \u0398 \u0012 \u03bbn d|U| \u0013\u0013 dk\u22121 nk\u22121|U|k, ||f||2 2 = ||g||2 2 = C2k\u22122(U) = (1 + o(1))|U|2k\u22122d2k\u22122 n2k\u22122 + \u0398 \u0012 \u03bb2k\u22124 d n|U|2 \u0013 , ||F||2 2 = ||G||2 2 = ||F \u2032||2 2 = ||G\u2032||2 2 = P2k\u22122(U) = (1 + o(1))d2k\u22122 n2k\u22122|U|2k\u22121. By applying Proposition 3.2 and the estimates above, we get that \f \f \f \fC2k(U) \u2212d2 n2Pk\u22121(U) \f \f \f \f \u2264\u03bb2C2k\u22122(U) + 2\u03bbd n2 P2k\u22122(U), and hence \f \f \f \fC2k(U) \u2212 \u0012 1 + \u0398 \u0012 \u03bbn d|U| \u0013\u0013 |U|2kd2k n2k \f \f \f \f = O \u0012|U|2k\u22122d2k\u22122\u03bb2 n2k\u22122 + \u03bb2k\u22122|U|2d n + \u03bb|U|2k\u22121d2k\u22121 n2k \u0013 . Since \u03bb|U|2k\u22121d2k\u22121 n2k and |U|2k\u22122d2k\u22122\u03bb2 n2k\u22122 are both o \u0010 \u03bbn d |U|2kd2k n2k \u0011 (the latter because of the assumption that \u03bbn d = o(|U|)), we are done. 3.5 Proof of Theorem 1.5 using the second counting lemma Proof of Theorem 1.5. Using the second counting lemma, we are able to prove Theorem 1.5 for all m \u22655, i.e. Cm(U) = |U|mdm nm + \u0398 \u0012\u03bb|U|m\u22121dm\u22121 nm\u22121 + \u03bbm\u22122 d n|U|2 \u0013 , but for m = 4, the result becomes slightly weaker, namely, C4(U) = O \u0012|U|4d4 n4 + \u03bb2|U|2d n \u0013 . We proceed by induction. Case 1: m = 2k. For m = 4, by Corollary 3.4 and Proposition 3.5, we have that \f \f \f \fC4(U) \u2212|U|4d4 n4 \u0012 1 + \u0398 \u0012 \u03bbn d|U| \u0013\u0013\f \f \f \f \u2264 r (1 + o(1))|U|2\u03bb2d n C4(U), (14) using that C2(U) = P2(U) and P2(U) = d|U|2 n (1 + o(1)) by the assumption that \u03bbn d = o(|U|). 26 \fHence we may set up a quadratic in p C4(U) to obtain C4(U) \u2264 \uf8eb \uf8ec \uf8ec \uf8ed q (1 + o(1))|U|2\u03bb2d n + r (1 + o(1))|U|2\u03bb2d n + 4 |U|4d4 n4 \u0010 1 + \u0398 \u0010 \u03bbn d|U| \u0011\u0011 2 \uf8f6 \uf8f7 \uf8f7 \uf8f8 2 = O \u0012|U|4d4 n4 + \u03bb2|U|2d n \u0013 . This gives the desired estimate for C4. If one wishes to have the main term |U|4d4 n4 instead of c |U|4d4 n4 , for some positive constant c, with this approach, then it can be pushed further as follows. Using the above upper bound for C4 and the estimate (14) gives us \f \f \f \fC4(U) \u2212|U|4d4 n4 \u0012 1 + \u0398 \u0012 \u03bbn d|U| \u0013\u0013\f \f \f \f = O  r |U|4\u03bb4d2 n2 + |U|6\u03bb2d5 n5 ! . This gives C4(U) = |U|4d4 n4 + \u0398 \u0012\u03bb2d|U|2 n + \u03bb|U|3d3 n3 + |U|3\u03bbd5/2 n5/2 \u0013 . Note that this gives the estimate (3) under the more restrictive condition that \u03bb n3/2 d3/2 = o(|U|). Assume the upper bound holds for all cycles of length at most m \u22121. We now show that it also holds for cycles of length m. Indeed, if m = 2k, then we can apply Corollary 3.4 to have C2k(U) \u2264d nP2k\u22121(U) + \u03bb (C2k(U)C2k\u22122(U))1/2 . Solving a quadratic in p C2k(U) gives C2k(U) \u2264 \uf8eb \uf8ed\u03bb p C2k\u22122(U) + q \u03bb2C2k\u22122(U) + 4 d nP2k\u22121(U) 2 \uf8f6 \uf8f8 2 Using Proposition 3.5 gives that C2k(U)\u2212|U|2kd2k n2k \u0012 1 + \u0398 \u0012 \u03bbn d||U| \u0013\u0013 = O   \u03bb2C2k\u22122(U) + \u03bb r \u03bb2(C2k\u22122(U))2 + d nC2k\u22122(U)P2k\u22121(U) ! . By the inductive hypothesis and the assumption that \u03bbn d = o(|U|), we have that \u03bb2C2k\u22122(U) = O \u0012\u03bb|U|2k\u22121d2k\u22121 n2k\u22121 + \u03bb2k\u22122|U|2d n \u0013 , 27 \fand hence we are done as long as \u03bb r d nC2k\u22122(U)P2k\u22121(U) = O \u0012\u03bb|U|2k\u22121d2k\u22121 n2k\u22121 + \u03bb2k\u22122|U|2d n \u0013 . By the inductive hypothesis and Proposition 3.5, we have \u03bb r d nC2k\u22122(U)P2k\u22121(U) = O  r \u03bb2|U|4k\u22122d4k\u22122 n4k\u22122 + \u03bb3|U|4k\u22123d4k\u22123 n4k\u22123 + \u03bb2k\u22122|U|2k+2d2k+1 n2k+1 ! . Since \u03bbn d = o(|U|) we have that \u03bb3|U|4k\u22123d4k\u22123 n4k\u22123 = o \u0010 \u03bb2|U|4k\u22122d4k\u22122 n4k\u22122 \u0011 . Therefore, because r \u03bb2|U|4k\u22122d4k\u22122 n4k\u22122 = \u03bb|U|2k\u22121d2k\u22121 n2k\u22121 , we are done as long as \u03bbk\u22121|U|k+1d(2k+1)/2 n(2k+1)/2 is small enough. If |U| \u2265\u03bbn(2k\u22123)/(2k\u22124) d(2k\u22123)/(2k\u22124) then \u03bbk\u22121|U|k+1d(2k+1)/2 n(2k+1)/2 \u2264\u03bb|U|2k\u22121d2k\u22121 n2k\u22121 . Otherwise, we have \u03bbk\u22121|U|k+1d(2k+1)/2 n(2k+1)/2 \u2264\u03bb2k\u22122|U|2d n , and the upper bound is complete. An analogous calculation gives the corresponding lower bound and we omit the details. Case 2: m = 2k + 1. This case follows directly from Corollary 3.4 and the case m = 2k above. 4 Proofs of Theorem 1.7 and Theorem 1.8 4.1 Technical lemmas To prove Theorems 1.7 and 1.8, we use the following results, which are direct consequences of the expander mixing lemma. The \ufb01rst result guarantees that vertex sets bigger than \u03bbn/d will have an edge of each color. Lemma 4.1. Let G be an (n, d, \u03bb)-colored graph with color set D, and A, B \u2282V (G) with |A| = |B| > \u03bbn d . Then for each color c \u2208D, there exists an edge uv of color c with u \u2208A and v \u2208B. In other words, every vertex set of size greater than \u03bbn d determines every color. Proof. For each color c in D, let Gc be the induced graph on c, then Gc is an (n, d, \u03bb)-graph. 28 \fApplying Lemma 2.5 with f(u) = \uf8f1 \uf8f2 \uf8f3 1, if u \u2208A 0, otherwise and g(v) = \uf8f1 \uf8f2 \uf8f3 1, if v \u2208B 0, otherwise, we have \u27e8f, Ag\u27e9= e(A, B) := |{(a, b) \u2208A \u00d7 B : ab \u2208E(G)}| . It is clear that E(f) = |A| n , E(g) = |B| n and \u2225f\u22252 = p |A|, \u2225g\u22252 = p |B|. Then we have \f \f \f \fe(A, B) \u2212d n|A||B| \f \f \f \f \u2264\u03bb p |A||B|. So e(A, B) \u2265d n|A||B| \u2212\u03bb p |A||B|. Since |A| = |B| > \u03bbn d , e(A, B) \u2265d n|A|2 \u2212\u03bb|A| \u2265|A| \u0012d n \u00b7 |A| \u2212\u03bb \u0013 > |A| \u0012d n \u00b7 \u03bbn d \u2212\u03bb \u0013 > 0. Which means that there exists at least one edge of color c between A and B. The next technical lemma uses the previous result to give an upper bound on the number of vertices with small degree of a given edge color. Lemma 4.2. Let G be an (n, d, \u03bb)-colored graph with color set D, and let U \u2282V (G) with |U| = r \u00b7 \u03bbn d . Then for any \ufb01xed color d \u2208D, s \u2208N, there are at most s \u00b7 \u03bbn d vertices of U for which each of them is incident with less than s edges colored by d. Proof. Let H be induced graph on color d. Consider the subgraph H\u2217of H generated by only those vertices of degree less than s, so H\u2217can be s-colorable. That is, we have a vertex partition into s independent sets. Using Lemma 4.1, an independent set in H (and thus in H\u2217) has size at most \u03bbn d . Otherwise, by Lemma 4.1, every vertex set of size greater than \u03bbn d determines every color, which means there exists two vertices connected by a d-color edge, contradicting the independence. As a result |V (H\u2217)| \u2264s \u00b7 \u03bbn d , proving the lemma. 29 \fThe next lemma develops this further by giving lower bounds on the number of disjoint copies of star graphs. Lemma 4.3. Let G be an (n, d, \u03bb)-colored graph with color set D, and let U \u2282V (G) with |U| = r \u00b7 \u03bbn d . Then the number of vertex disjoint copies of the nonempty star graph K1,m with any \ufb01xed edge-coloring from D is at least r\u2212m m+1 \u00b7 \u03bbn d . Proof. Let T be the maximal set of copies of K1,m in U, and H be the union of all copies in T. Then U \u2212H will have no copies of K1,m. Suppose the set of color of K1,m is {c1, c2, . . . , ct} with multiplicities {m1, m2, . . . , mt}. Using Lemma 4.2, for each i there are at most mi \u00b7 \u03bbn d vertices that are incident with fewer than mi edges colored by ci. Summing over i we get that there are at most t X i=1 mi \u00b7 \u03bbn d = m \u00b7 \u03bbn d vertices of U \u2212H which are not colored ci from at least mi other vertices of U \u2212H for every i. If vertex v \u2208U \u2212H is incident with at least mi edges color ci for every i, then v is the singleton bipartition set of an instance of K1,m. Thus |U \u2212H| \u2264m \u00b7 \u03bbn d . By disjointness |T| = |H| m + 1 \u2265r \u00b7 \u03bbn d \u2212m \u00b7 \u03bbn d m + 1 = r \u2212m m + 1 \u00b7 \u03bbn d as required. Our \ufb01nal technical lemma is a simple application of Lemma 4.2 that gives a lower bound on the number of disjoint edges of a given color in a vertex set. Lemma 4.4. Let G be an (n, d, \u03bb)-colored graph with color set D, and let U \u2282V (G) with |U| \u22652\u03bbn d . Then for each color c \u2208D, the number of disjoint c colored edges in U is at least |U| 2 \u2212\u03bbn d . Proof. We partition the vertex set of U into two sets, A and B, such that |A| = |B| = |U| 2 . Choose as large a matching of color c as possible between, say, A\u2032 \u2286A and B\u2032 \u2286B. We have that the two sets A \\ A\u2032 and B \\ B\u2032 both have size at most \u03bbn d . Otherwise, by Lemma 4.1 we could increase the size of our matching. As a result, the number of disjoint c colored edges in U is at least |A \u2032| = |B \u2032| \u2265|U| 2 \u2212\u03bbn d as required. 30 \f4.2 Proof of Theorem 1.7 The proof proceeds by strong induction on the number of edges in T. If T contains no edges the theorem is clearly true; if T is a star graph K1,m, then \u03c3(G) = m + 1 and the theorem is Lemma 4.3. Now assume T is not a star graph. Let T \u2032 be the graph produced by deleting all leaves of T. Since T is not a star graph, T \u2032 is a tree which has at least two leaves, we can choose v be a leaf of T \u2032 such that there exists another leaf of T \u2032, say w, such that degT v \u2264degT w. Suppose the set of leaves of T connected to v is {v1, v2, . . . , vy}. De\ufb01ne the graph T \u2217 to be T \\ {v1, v2, . . . , vy}. By construction, T \u2217is a tree with fewer edges than T and \u03c3(T) = \u03c3(T \u2217) \u00b7 (y + 1). By the inductive hypothesis we have the number of disjoint copies of T \u2217in U denoted by CT \u2217is at least \u0010 r \u03c3(T \u2217) \u22121 \u0011 \u00b7 \u03bbn d . We are building our tree T out of stars instead of edges. Let W be the set of copies of v in U. By disjointness |W| = |CT \u2217|. Let K1,y be the star graph generated by {v, v1, v2, . . . , vy} where the root is v. Using Lemma 4.3, there exists at least |W|/ \u03bbn d \u2212y y+1 \u00b7 \u03bbn d disjoint copies of K1,y in W. For each copy of K1,y we can build our tree T by adding the copies of T \u2217that correspond to v. These are disjoint copies of T because of the disjointness of T \u2217and the disjointness of K1,y. So there are at least |W|/ \u03bbn d \u2212y y + 1 \u00b7 \u03bbn d = |CT \u2217|/ \u03bbn d \u2212y y + 1 \u00b7 \u03bbn d \u2265 \u0010 r \u03c3(T \u2217) \u22121 \u0011 \u2212y y + 1 \u00b7 \u03bbn d = \u0012 r (y + 1)\u03c3(T \u2217) \u22121 \u0013 \u00b7 \u03bbn d = \u0012 r \u03c3(T) \u22121 \u0013 \u00b7 \u03bbn d disjoint copies of T as required. 4.3 Proof of Theorem 1.8 The proof proceeds by induction on the number of edges on T. If T contains no edges, the theorem is clearly true. If T is an edge, then |V (T)| = 2, the theorem is Lemma 4.4. So assume T is a tree with m vertices. Consider the subgraph T \u2217of T produced by deleting one leaf on vertex x. Let\u2019s say the edge we are just removing has color c. By construction T \u2217 is a tree with fewer edges than T, say m\u22121. By inductive hypothesis we have the collection of disjoint copies of T \u2217in U is at least |U| m\u22121 \u2212\u03bbn d . Choose |U| m copies of them arbitrarily and let this set of vertices be called S. This is possible since |U| \u2265m(m \u22121) \u03bbn d . 31 \fSo S has size (m \u22121) \u00b7 |U| m . Now in these copies of T \u2217, denote by A the set copies of x to which we will be trying to add an edge of color c, so |A| = |U| m . Let B = U \\ S so |B| = |U| \u2212(m \u22121) \u00b7 |U| m = |U| m . Choose as large of a matching color c as possible between, say, A\u2032 \u2286A and B\u2032 \u2286B, each matching creates a copy of T. Let C and D be the sets of vertices in A \\ A\u2032 and B \\ B\u2032 respectively. Then we have that |C| = |D| \u2264\u03bbn d . Otherwise using Lemma 4.1 we can \ufb01nd at least one c colored edge between C and D, which would increase the size of our matching. So the number of disjoint copies of T is |A\u2032| = |A| \u2212|C| \u2265|U| m \u2212\u03bbn d , as required. 5 Acknowledgements T. Pham would like to thank to the VIASM for the hospitality and for the excellent working conditions. M. Tait was partially supported by National Science Foundation grant DMS2011553 and a Villanova University Summer Grant."
+                },
+                {
+                    "url": "http://arxiv.org/abs/2103.11420v2",
+                    "title": "An inverse-type problem for cycles in local Cayley distance graphs",
+                    "abstract": "Let $E$ be a proper symmetric subset of $S^{d-1}$, and\n$C_{\\mathbb{F}_q^d}(E)$ be the Cayley graph with the vertex set\n$\\mathbb{F}_q^d$, and two vertices $x$ and $y$ are connected by an edge if\n$x-y\\in E$. Let $k\\ge 2$ be a positive integer. We show that for any $\\alpha\\in\n(0, 1)$, there exists $q(\\alpha, k)$ large enough such that if $E\\subset\nS^{d-1}\\subset \\mathbb{F}_q^d$ with $|E|\\ge \\alpha q^{d-1}$ and $q\\ge q(\\alpha,\nk)$, then for each vertex $v$, there are at least $c(\\alpha,\nk)q^{\\frac{(2k-1)d-4k}{2}}$ cycles of length $2k$ with distinct vertices in\n$C_{\\mathbb{F}_q^d}(E)$ containing $v$. This result is the inverse version of a\nrecent result due to Iosevich, Jardine, and McDonald (2021).",
+                    "authors": "Thang Pham",
+                    "published": "2021-03-21",
+                    "updated": "2021-05-08",
+                    "primary_cat": "math.CO",
+                    "cats": [
+                        "math.CO"
+                    ],
+                    "main_content": "Introduction Let G be an abelian \ufb01nite group and a symmetric set E \u2282G. The Cayley graph CG(E) is de\ufb01ned as the graph with the vertex set V = G, and there is an edge from x to y if y \u2212x \u2208E. Let Fq be a \ufb01nite \ufb01eld of order q, where q is a prime power. In this paper, we consider G being the whole vector space Fd q. We have CFd q(E) is a regular graph of degree |E| with qd vertices. It is well\u2013known in the literature that eigenvalues of CFd q(E) are of the form \u03bbm := P x\u2208E \u03c7(x \u00b7 m) = b E(m), m \u2208Fd q, where \u03c7 is the principle additive character of Fq. De\ufb01ne \u00b5 := maxm\u0338=(0,0,...,0) |\u03bbm|. This quantity is referred as the second largest eigenvalue of CFd q(E). We call a graph (n, d, \u03bb)-graph if it has n vertices, the degree of each vertex is d, and the second largest eigenvalue is at most \u03bb. When E = Sd\u22121, the unit sphere in Fd q, we recall a result from a paper of Iosevich and Rudnev [13, Lemma 5.1] that \u00b5 = (1 + o(1))q d\u22121 2 . Thus, the graph CFd q(Sd\u22121) is a (n, d, \u03bb)-graph with n = qd, d = |Sd\u22121|, and \u03bb = (1 + o(1))q d\u22121 2 . In a (n, d, \u03bb)-graph, we know from [14, Theorem 4.10] that any large subset of vertices contains the correct number of copies of any \ufb01xed sparse graph. More \u2217Theory of Combinatorial Algorithms Group, ETH Zurich, Switzerland. Email: phamanhthang.vnu@gmail.com 1 \fprecisely, let H be a \ufb01xed graph with r edges, s vertices, and maximal degree \u2206, then any subset A \u2282Fd q of m vertices with m \u226b\u03bb \u0000 n d \u0001\u2206contains about ms(d/n)r copies of H. The condition m \u226b\u03bb \u0000n d \u0001\u2206can be improved when H is of some speci\ufb01c con\ufb01guration, for instance, paths, stars, complete graphs [4, 5, 7, 9, 12, 17, 18]. 1 In the inverse setting, we let E be a proper subset of Sd\u22121 and A = Fd q, the question is to \ufb01nd conditions on E such that the graph CFd q(E), which will be called local Cayley distance graphs, contains at least one copy of H. The main purpose of this paper is to study the inverse version of a recent result due to Iosevich, Jardine, and McDonald in [12] on the distribution of cycles. We start by stating their result. Theorem 1.1 (Iosevich-Jardine-McDonald, [12]). Let A be a set in Fd q. Suppose that |A| \u226bq d+2 2 , then for any positive integer \u2113\u22653, the number of cycles of length \u2113in CFd q(Sd\u22121) with vertices in A is (1 + o(1))|A|\u2113q\u2212\u2113. In addition, when \u2113is large, then the exponent d+2 2 can be improved, namely, the condition |A| \u2265 \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 q 1 2(d+2\u2212\u2113\u22124 \u2113\u22122+\u03b4), if \u2113\u22654 even q 1 2(d+2\u2212\u2113\u22123 \u2113\u22121+\u03b4), if \u2113\u22653 odd , where 0 < \u03b4 \u226a1 \u21132, would be enough. The following is our main result. Theorem 1.2. Let d, k \u2208N with d \u22654k + 2, \u03b1 \u2208(0, 1) and q \u2265q(\u03b1, k). For a symmetric set E \u2282Sd\u22121 \u2282Fd q with |E| \u2265\u03b1qd\u22121, the number of cycles of length 2k in CFd q(E) with distinct vertices passing through each vertex of CFd q(E) is at least c(\u03b1, k)q (2k\u22121)d\u22124k 2 . To prove Theorem 1.2, several serious challenges arise, and the most di\ufb03culty comes from the fact the graph CFd q(E) is not a pseudo-random graph, namely, the second eigenvalue \u00b5 is arbitrary close to the graph degree when q is large enough. Proposition 1.3. For any 1 \u2264m \u226aqd\u22121 and \u01eb > 0 with 1/\u01eb \u2208Z. Let q = p 1 \u01eb . There exists E \u2282Sd\u22121 such that |E| = m and \u00b5 \u2265|E| 2q\u01eb . In addition, if |E + E| \u223c|E|, then we have \u00b5 \u223c \u03bb(0,...,0) = |E|. Hence, it is not possible to apply techniques of pseudo-random graphs to prove such a result as Theorem 1.2. Our main ingredient is a recent Ramsey-type result on the number of congruence copies of 2k-spherical con\ufb01gurations spanning 2k\u22122 dimensions due to Lyall, Magyar, and Parshall in [15], which has been derived by using a generalized von-Neumann type inequality [15, Proposition 1We use the following notations: X \u226aY means that there exists some absolute constant C > 0 such that X \u2264CY , X \u223cY means that X \u226aY \u226aX, X = o(Y ) means that limq\u2192\u221eX/Y = 0. 2 \f6] and an inverse theorem [15, Proposition 7]. It seems di\ufb03cult to extend the approach of Theorem 1.2 for other subgraphs H. When H is a k-simplex, say k = 2 for simplicity, the inverse problem asks for conditions on three given proper subsets E1, E2, E3 of Sd\u22121 such that there are three vertices x, y, z \u2208Fd q such that x \u2212y \u2208 E1, y \u2212z \u2208E2, z \u2212x \u2208E3. Note that E1, E2, E3 can also be assumed to be subsets of spheres with di\ufb00erent radii. We believe that \ufb01nding a non-trivial solution of this problem would be much di\ufb03cult compared to the original one. When E = Sd\u22121, giving a lower bound on the number of cycles in CFd q(E) is much easier, since, as mentioned earlier, CFd q(Sd\u22121) is a pseudo-random graph with the second eigenvalue \u00b5 \u223c p |Sd\u22121|. In the next proposition, we provide an improvement of Theorem 1.1 in terms of the lower bound on the number of cycles of even length. Proposition 1.4. Suppose E = Sd\u22121, then the number of cycles of length 2k in CFd q(Sd\u22121) is (1+o(1))|Sd\u22121|2k\u22121qd\u22121. In addition, for any set A \u2282Fd q with |A| \u226bmin{q d+1 2 , q k k\u22121 }, the number of cycles of length 2k in CFd q(E) with vertices in A is at least q\u22122k|A|2k. Based on Proposition 1.4 and in the spirit of Theorem 1.1, we conjecture that for any set A \u2282Fd q with |A| \u226bmin{q d+1 2 , q k k\u22121}, the number of cycles of length 2k in CFd q(Sd\u22121) with vertices in A is equal to (1 + o(1))q\u22122k|A|2k. 2 Preliminaries Let \u03c7: Fq \u2192S1 be the canonical additive character. For example, if q is a prime number, then \u03c7(t) = e 2\u03c0it q , if q = pn, then we set \u03c7(t) = e 2\u03c0iTr(t) q , where Tr: Fq \u2192Fq is the trace function de\ufb01ned by Tr(x) := x + xp + \u00b7 \u00b7 \u00b7 + xpn\u22121. We recall the orthogonal property of \u03c7: for any x \u2208Fd q, d \u22651, X m\u2208Fd q \u03c7(x \u00b7 m) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 0 if x \u0338= (0, . . . , 0) qd if x = (0, . . . , 0) , where x \u00b7 m = x1m1 + \u00b7 \u00b7 \u00b7 + xdmd. For any x \u2208Fd q, through this paper, we de\ufb01ne ||x|| = x2 1 + \u00b7 \u00b7 \u00b7 + x2 d. Given a set E \u2282Fd q, we identify E with its indicator function 1E. The Fourier transform of E is de\ufb01ned by b E(m) := X x\u2208Fd q E(x)\u03c7(\u2212x \u00b7 m). 3 \fLet E be a set in Fd q, and k be a positive integer. The k\u2013additive energy of E, denoted by Tk(E), is de\ufb01ned by Tk(E) := # n (a1, . . . , ak, b1 . . . , bk) \u2208E2k : a1 + \u00b7 \u00b7 \u00b7 + ak = b1 + \u00b7 \u00b7 \u00b7 + bk o . We call such a tuple (a1, . . . , ak, b1, . . . , bk) k-energy tuple. A k-energy tuple (a1, . . . , ak, b1, . . . , bk) \u2208 \u0000Fd q \u00012k is called good if for any two sets of indices I, J \u2282{1, . . . , k}, we have P i\u2208I ai \u2212P j\u2208J bj \u0338= 0. We denote the number of good k-energy tuples with vertices in E by T good k (E). In the next lemma, we show that for every vertex v \u2208Fd q, the number of cycles of length 2k with distinct vertices going through v is at least T good k (E). Lemma 2.1. For any k \u22652 and any v \u2208Fd q, the number of cycles of length 2k in CFd q(E) with distinct vertices going through v is at least T good k (E). Proof. For each good k-energy tuple (a1, . . . , ak, b1, . . . , bk) \u2208E2k, we consider the following cycle of length 2k in CFd q(E) : v, v + a1, v + a1 + a2, . . . , v + a1 + \u00b7 \u00b7 \u00b7 + ak, v + k X i=1 ai \u2212b1, \u00b7 \u00b7 \u00b7 , v + k X i=1 ai \u2212 k\u22121 X i=1 bi. We observe that in this cycle, each vertex appears only one time since the k-energy tuple is good. So, for each vertex v, there are at least T good k (E) cycles with distinct vertices passing through v. We also recall the well\u2013known Expanding mixing lemma for regular graphs. We refer the reader to [10, 14] for proofs. Lemma 2.2. Let G be a regular graph with n vertices of degree d. Suppose that the second eigenvalue of G is at most \u00b5, then for any two vertex sets U and W in G, the number of edges between U and W, denoted by e(U, W), satis\ufb01es \f \f \f \fe(U, W) \u2212d|U||W| n \f \f \f \f \u2264\u00b5|U|1/2|W|1/2. When U and W are multi-sets, we also have \f \f \f \fe(U, W) \u2212d|U||W| n \f \f \f \f \u2264\u00b5 \uf8eb \uf8edX u\u2208U m(u)2 \uf8f6 \uf8f8 1/2 \u00b7 \uf8eb \uf8edX w\u2208W m(w)2 \uf8f6 \uf8f8 1/2 , 4 \fwhere X is the set of distinct elements in X, and m(x) is the multiplicity of x. 3 Proof of Theorem 1.2 Theorem 1.2 follows directly from Lemma 2.1 and the following lower bound for T good k (E). Theorem 3.1. Suppose E satis\ufb01es assumptions of Theorem 1.2, we have T good k (E) \u2265c(\u03b1, k)q (2k\u22121)d\u22124k 2 . In the rest of this section, we focus on proving Theorem 3.1. For each j \u0338= 0, let Sd\u22121 j (x) be the sphere centered at x \u2208Fd q of radius j. For the sake of simplicity, we write Sd\u22121 j for Sd\u22121 j (0, . . . , 0), and Sd\u22121 for Sd\u22121 1 (0, . . . , 0). De\ufb01nition 3.2. Let X \u2282Fd q be a con\ufb01guration. We say that X is spherical if X \u2282Sd\u22121 1 (x) for some x \u2208Fd q. If dim(Span(X \u2212X)) = k, then we say X spans k dimensions. The following result is our key ingredient in the proof of Theorem 3.1. Theorem 3.3 (Lyall-Magyar-Parshall, [15]). Let d, k \u2208N with d \u22652k + 6, \u03b1 \u2208(0, 1) and q \u2265 q(\u03b1, k). For E \u2282Sd\u22121 with |E| \u2265\u03b1qd\u22121, then E contains at least c(\u03b1, k)q (k+1)d\u2212(k+1)(k+2) 2 isometric copies of every non-degenerate (k + 2)-point spherical con\ufb01guration spanning k dimensions. This theorem says that for any \u03b1 \u2208(0, 1) and any \ufb01xed non-degenerate (k + 2)-point spherical con\ufb01guration X spanning k dimensions, there exists q0 = q0(\u03b1, k) which is large enough, such that for any E \u2282Sd\u22121 \u2282Fd q with |E| \u2265\u03b1qd\u22121 and q \u2265q0, E contains many isometric copies of X. More precisely, let X = {0, v1, . . . , vk, a1v1 + \u00b7 \u00b7 \u00b7 + akvk}, where 0 = (0, . . . , 0), v1, . . . , vk \u2208Fd q are linearly independent vectors, and a1, . . . , ak \u2208Fq, be a non-degenerate spherical con\ufb01guration of k + 2 points in Fd q that spans a k-dimensional vector space. By non-degenerate, we meant that {0, v1, . . . , vk} form a k\u2013simplex with all non-zero sidelengths. Assume that E \u2282Sd\u22121 satisfying the conditions of Theorem 3.3, then E contains at least c(\u03b1, k)q (k+1)d\u2212(k+1)(k+2) 2 copies of X of the form X\u2032 = {x0, x0 + x1, . . . , x0 + xk, x0 + a1x1 + \u00b7 \u00b7 \u00b7 + akxk}, with x1, . . . , xk linearly independent such that xi \u00b7 xj = vi \u00b7 vj for 1 \u2264i \u2264j \u2264k. We recall that two con\ufb01gurations X and X\u2032 in Sd\u22121 are said to be in the same congruence class if 5 \fthere exists g \u2208O(d, Fq), the orthogonal group in Fd q, such that g(X) = X\u2032. Let Q be the set of distinct congruence classes of spherical con\ufb01gurations X of the form X = {x0, x0 + x1, x0 + x2, . . . , x0 + x2k\u22122, x0 + 2k\u22122 X i=1 (\u22121)i+1(x + xi)}, satisfying \u2022 {x1, . . . , x2k\u22122} are linearly independent. \u2022 ||xi \u2212xj|| \u0338= 0, ||xi|| \u0338= 0 for all 1 \u2264i \u0338= j \u22642k \u22122. \u2022 X forms a good k-energy tuple. We note that vectors in X \u2208Q form a k-energy tuple since x0 + (x0 + x1) + (x0 + x3) + \u00b7 \u00b7 \u00b7 + (x0 + x2k\u22123) = (x0 + x2) + (x0 + x4) + \u00b7 \u00b7 \u00b7 + (x0 + x2k\u22122) + u, where u = x0 + P2k\u22122 i=1 (\u22121)i+1(x0 + xi). For each X \u2208Q, let N(X) be the number of congruent copies of X in E. Set N(Q) = P X\u2208Q N(X). The next lemma gives us a lower bound for T good k (E). Lemma 3.4. Suppose E satis\ufb01es assumptions of Theorem 1.2, we have T good k (E) \u2265N(Q). (1) Proof. Let X = {x0, x0 + x1, x0 + x2, . . . , x0 + x2k\u22122, x0 + 2k\u22122 X i=1 (\u22121)i+1(x0 + xi)} \u2208Q, and set u = x0 + P2k\u22122 i=1 (\u22121)i+1(x0 + xi), then we have x0 + (x0 + x1) + (x0 + x3) + \u00b7 \u00b7 \u00b7 + (x0 + x2k\u22123) = (x0 + x2) + (x0 + x4) + \u00b7 \u00b7 \u00b7 + (x0 + x2k\u22122) + u, which provides a good k-energy tuple. Notice that x0 + xi \u0338= x0 + xj for all pairs (i, j), and u \u0338= x0, x0 + xi for all i. Since the additive energy is invariant under the action of orthogonal matrices, we have N(X) good k-energy tuples in E. Summing over all X, we have N(Q) good k-energy tuples in E. In the form of Lemma 3.4, in order to complete the proof of Theorem 3.1, we have to \ufb01nd a lower 6 \fbound for N(Q), which will be followed by a lower bound of |Q| and Theorem 3.3. The following proposition plays an important role for this step. Proposition 3.5. For d \u2265max{2k \u22122, 4} and k \u22652, we have |Q| \u226bq2k2\u22123k. With Proposition 3.5 in hand, we derive the following corollary. Corollary 3.6. Let d, k \u2208N with d \u22654k + 2, \u03b1 \u2208(0, 1) and q \u2265q(\u03b1, k). Let E \u2282Sd\u22121 \u2282Fd q with |E| \u2265\u03b1qd\u22121. We have N(Q) \u2265c(\u03b1, k)q (2k\u22121)d\u22124k 2 . Proof. For each con\ufb01guration in Q, we know from Theorem 3.3 that the number of its copies in E is at least c(\u03b1, k)q (2k\u22121)d\u2212(2k\u22121)(2k) 2 . Taking the sum over all possible q2k2\u22123k congruence classes, the lemma follows. Combining Lemma 3.4 and Corollary 3.6, Theorem 3.1 is proved. 3.1 Proof of Proposition 3.5 We now turn our attention to the Proposition 3.5. The proof of Proposition 3.5 is quite complicated, which combines the usual Cauchy-Schwarz argument and the claim that most k-energy tuples in Sd\u22121 are 2k-spherical con\ufb01gurations spanning (2k \u22122) dimensions. We \ufb01rst start with some technical lemmas. Lemma 3.7 (Lemma 4.5, [11]). For any E \u2286Sd\u22121, and k \u22652, we have \f \f \f \fTk(E) \u2212|E|2k\u22121 q \f \f \f \f \u2264q d\u22121 2 T 1/2 k T 1/2 k\u22121, where T1(E) = |E|. Corollary 3.8. For k, d \u22652, we have Tk(Sd\u22121) = (1 + o(1))|Sd\u22121|2k\u22121 q . Proof. We prove by induction on k. For k = 2, we apply Lemma 3.7 to obtain \f \f \f \fT2(Sd\u22121) \u2212|Sd\u22121|3 q \f \f \f \f \u2264q d\u22121 2 \u00b7 T 1/2 2 |Sd\u22121|1/2. 7 \fUsing the fact that |Sd\u22121| \u223cqd\u22121 and set x = p T2(Sd\u22121), we have x2 \u2265c1q3d\u22124 \u2212c2qd\u22121x, and x2 \u2264c1q3d\u22124 + c2qd\u22121x, for some positive constants c1 and c2. Solving these equations gives us x \u226bq 3d\u22124 2 and x \u226aq 3d\u22124 2 , respectively. Thus, the base case is proved. Suppose that the claim holds for any k \u22121 \u22652, we now show that it also holds for the case k. Indeed, set x = p Tk(Sd\u22121), applying Lemma 3.7 and the inductive hypothesis, we have x2 \u2212q d\u22121 2 |Sd\u22121| 2k\u22123 2 x \u2212|Sd\u22121|2k\u22121 q \u22640, x2 + q d\u22121 2 |Sd\u22121| 2k\u22123 2 x \u2212|Sd\u22121|2k\u22121 q \u22650. Solving these inequalities will give us x = (1 + o(1)) \u0012|Sd\u22121|2k\u22121 q \u00131/2 . This completes the proof of the corollary. Lemma 3.9. For d > n \u22652, let L be the number of tuples (v0, . . . , vn) \u2208(Sd\u22121)n+1 such that vi \u2212 v0 \u2208{a1(v1\u2212v0)+\u00b7 \u00b7 \u00b7+ai\u22121(vi\u22121\u2212v0)+ai+1(vi+1\u2212v0)+\u00b7 \u00b7 \u00b7+an(vn\u2212v0): a1, . . . , ai\u22121, ai+1, . . . , an \u0338= 0} for some 1 \u2264i \u2264n. We have L \u226a|Sd\u22121|n+1 q2 . Proof. Without loss of generality, we count the number of such tuples with i = n. Let \u03c7 be the principle additive characteristic of Fq. Using the orthogonality of \u03c7, one has L \u22641 qd X s\u2208Fd q X v0,...,vn\u2208Sd\u22121 X a1,...,an\u22121\u2208F\u2217 q \u03c7 \u0012 s \u00b7 \u0012 (vn \u2212v0) \u2212a1(v1 \u2212v0) \u2212\u00b7 \u00b7 \u00b7 \u2212an\u22121(vn\u22121 \u2212v0) \u0013\u0013 = |Sd\u22121|n+1 qd\u2212n+1 + 1 qd X s\u0338=0 X v0,...,vn\u2208Sd\u22121 X a1,...,an\u22121\u2208F\u2217 q \u03c7 \u0012 s \u00b7 \u0012 (vn \u2212v0) \u2212a1(v1 \u2212v0) \u2212\u00b7 \u00b7 \u00b7 \u2212an\u22121(vn\u22121 \u2212v0) \u0013\u0013 = |Sd\u22121|n+1 qd\u2212n+1 + 1 qd X s\u0338=0 X a1,...,an\u22121\u2208F\u2217 q [ Sd\u22121(a1s) \u00b7 \u00b7 \u00b7 [ Sd\u22121(an\u22121s)[ Sd\u22121(s)[ Sd\u22121(s(1 \u2212a1 \u2212\u00b7 \u00b7 \u00b7 \u2212an\u22121)), where b S(m) = P x\u2208Fd q S(x)\u03c7(\u2212x \u00b7 m). We now recall from [13, Lemma 5.1] that |[ Sd\u22121(m)| \u226aq d\u22121 2 for m \u0338= 0 and b S(0) = |Sd\u22121| \u223cqd\u22121. We now partition the sum P a1,...,an\u22121\u2208F\u2217 q into two subsummands P a1+\u00b7\u00b7\u00b7+an\u22121\u0338=1 and P a1+\u00b7\u00b7\u00b7+an\u22121=1. 8 \fTherefore, X a1+\u00b7\u00b7\u00b7+an\u22121\u0338=1 [ Sd\u22121(a1s) \u00b7 \u00b7 \u00b7 [ Sd\u22121(an\u22121s)[ Sd\u22121(s)[ Sd\u22121(s(1 \u2212a1 \u2212\u00b7 \u00b7 \u00b7 \u2212an\u22121)) \u226aq (d\u22121)(n+1) 2 \u00b7 qn\u22121, and X a1+\u00b7\u00b7\u00b7+an\u22121=1 [ Sd\u22121(a1s) \u00b7 \u00b7 \u00b7 [ Sd\u22121(an\u22121s)[ Sd\u22121(s)[ Sd\u22121(s(1 \u2212a1 \u2212\u00b7 \u00b7 \u00b7 \u2212an\u22121)) \u226aq (d\u22121)(n) 2 \u00b7 qd\u22121 \u00b7 qn\u22122. These upper bounds are at most |Sd\u22121|n+1 qd\u2212n+1 when d > n and n \u22652. In other words, L \u226a|Sd\u22121|n+1 q2 . Lemma 3.10. Suppose that d > 2k \u22122 and k \u22652. The number of tuples {x0, x0 + x1, . . . , x0 + x2k\u22122, x0 + P2k\u22122 i=1 (\u22121)i+1(x0 + xi)} in (Sd\u22121)2k such that x0 + (x0 + x1) + (x0 + x3) + \u00b7 \u00b7 \u00b7 + (x0 + x2k\u22123) = (x0 + x2) + (x0 + x4) + \u00b7 \u00b7 \u00b7 + (x0 + x2k\u22122) + u, where u = x0 + P2k\u22122 i=1 (\u22121)i+1(x0 + xi), and xi \u2208Span(x1, . . . , xi\u22121, xi+1, . . . , x2k\u22122) for some 1 \u2264i \u22642k \u22122 is o(Tk(Sd\u22121)) . Proof. Applying Lemma 3.9 for the family of vectors {x0, x0+x1, . . . , x0+x2k\u22122} or its sub-families, we know that there are at most |Sd\u22121|2k\u22121 q2 such tuples whenever d > 2k \u22122 and k \u22652. We also know from Corollary 3.8 that Tk(Sd\u22121) = (1 + o(1))|Sd\u22121|2k\u22121 q . Thus, the lemma follows from the fact that |Sd\u22121|2k\u22121 q2 = o \u0012|Sd\u22121|2k\u22121 q \u0013 . We are ready to give a proof of Proposition 3.5. Proof of Proposition 3.5. For any k-energy tuple (a1, . . . , ak, b1, . . . , bk) \u2208Sd\u22121, i.e. a1 + \u00b7 \u00b7 \u00b7 + ak = b1 + \u00b7 \u00b7 \u00b7 + bk, (2) we set ai = a1 + xi for 2 \u2264i \u2264k, and bi = a1 + yi for 1 \u2264i \u2264k. 9 \fWe \ufb01rst show that most of all tuples (a1, . . . , ak, b1, . . . , bk) satisfying (2) will have the following properties a. {x2, . . . , xk, y1, . . . , yk} are linearly independent. b. ||xi \u2212xj|| \u0338= 0, ||yi \u2212yj|| \u0338= 0 for all pairs i \u0338= j, and ||xi \u2212yj|| \u0338= 0, ||xi|| \u0338= 0, ||yj|| \u0338= 0 for all pairs i, j. c. For any I, J \u2286{1, . . . , k}, we have P i\u2208I ai \u2212P j\u2208J bj \u0338= 0. Indeed, let T dep k (Sd\u22121), T 0 k (Sd\u22121), T bad k (Sd\u22121) be the number of k-energy tuples not satisfying (a), (b), and (c), respectively. We will prove that T dep k (Sd\u22121), T 0 k (Sd\u22121), T bad k (Sd\u22121) = o(Tk(Sd\u22121)). Bounding T dep k : By Lemma 3.10, we have T dep k = o(Tk(Sd\u22121)). Bounding T 0 k : It follows from our setting that ||xi \u2212xj|| = ||ai \u2212aj|| and ||xi \u2212yj|| = ||ai \u2212bj||. Hence, it is su\ufb03cient to count tuples with ||ai \u2212aj|| = 0 for some 1 \u2264i \u0338= j \u2264k. The other cases can be treated in the same way. Without loss of generality, we assume that ||a1 \u2212a2|| = 0, which is equivalent with ||x2|| = 0. Let U be the multi-set de\ufb01ned by U := {a1 + \u00b7 \u00b7 \u00b7 + ak : ai \u2208Sd\u22121, ||a1 \u2212a2|| = 0}. Let W be the multi-set de\ufb01ned by W := {b1 + \u00b7 \u00b7 \u00b7 + bk\u22121 : bi \u2208Sd\u22121}. Let e(U, W) be the number of pairs (u, w) \u2208U \u00d7 W such that u \u2212w \u2208Sd\u22121. Applying Lemma 2.2 for the graph CFd q(Sd\u22121), we have e(U, W) \u2264|U||W| q + q d\u22121 2 \uf8eb \uf8edX u\u2208U m(u)2 \uf8f6 \uf8f8 1/2 \u00b7 \uf8eb \uf8edX w\u2208W m(w)2 \uf8f6 \uf8f8 1/2 , where m(u), m(w) are the multiplicities of u and w in U and W, respectively. We know from [8] that for any two sets X, Y \u2286Sd\u22121, the number of pairs (x, y) \u2208X \u00d7Y such that ||x \u2212y|| = 0 is at most |X||Y | q + q d 2 |X|1/2|Y |1/2. So with X = Y = Sd\u22121, we obtain |U| \u2264|Sd\u22121|k q . It is clear that |W| = |Sd\u22121|k\u22121. 10 \fOn the other hand, it is not hard to see that X u m(u)2 \u2264Tk(Sd\u22121), X w m(w)2 \u2264Tk\u22121(Sd\u22121). Using Corollary 3.8, one has e(U, W) \u2264|Sd\u22121|2k\u22121 q2 + q d\u22121 2 \u00b7 |Sd\u22121| 2k\u22121 2 q1/2 \u00b7 |Sd\u22121| 2k\u22123 2 q1/2 \u226a|Sd\u22121|2k\u22121 q2 . On the other hand, e(U, W) equals to the number of tuples satisfying (2) with ||a1 \u2212a2|| = 0. In other words, T 0 k \u226a|Sd\u22121|2k\u22121 q2 = o(Tk(Sd\u22121)). Bounding T bad k : Let I and J be two subsets of {1, . . . , k}. Assume that |I| = |J| = m. The case |I| \u0338= |J| is treated in the same way. Without loss of generality, we assume that I = J = {1, . . . , m}. We now count the number of k-energy tuples (a1, . . . , ak, b1, . . . , bk) \u2208(Sd\u22121)2k such that a1 + \u00b7 \u00b7 \u00b7 + am \u2212b1 \u2212\u00b7 \u00b7 \u00b7 \u2212bm = 0. This implies that am+1 + \u00b7 \u00b7 \u00b7 + ak \u2212bm+1 \u2212\u00b7 \u00b7 \u00b7 \u2212bk = 0. We now show that the number of tuples (a1, . . . , am, b1, . . . , bm) \u2208 \u0000Sd\u22121\u00012m such that a1 + \u00b7 \u00b7 \u00b7 + am \u2212b1 \u2212\u00b7 \u00b7 \u00b7 \u2212bm = 0 is at most \u226a|Sd\u22121|2m\u22121 q . Indeed, using the same argument as in bounding T 0 k , let U \u2032, W \u2032 be multi-sets de\ufb01ned by U \u2032 := {a1 + \u00b7 \u00b7 \u00b7 + am : ai \u2208Sd\u22121}, W = {b1 + \u00b7 \u00b7 \u00b7 + bm\u22121 : bi \u2208Sd\u22121}. The number of such tuples is bounded by e(U \u2032, W \u2032) in the graph CFd q(Sd\u22121). As before, we also have X u\u2208U\u2032 m(u)2 = Tm(Sd\u22121), X w\u2208W \u2032 m(w) = Tm\u22121(Sd\u22121). Using Lemma 2.2 and Lemma 3.7, we have e(U \u2032, W \u2032) \u226a|Sd\u22121|2m\u22121 q + q d\u22121 2 \u00b7 |Sd\u22121|2m\u22122 q \u226a|Sd\u22121|2m\u22121 q . Similarly, the number of tuples (am+1, . . . , ak, bm+1, . . . , bk) \u2208Sd\u22121 such that am+1 + \u00b7 \u00b7 \u00b7 + ak \u2212 bm+1 \u2212\u00b7 \u00b7 \u00b7 \u2212bk = 0 is at most \u226a|Sd\u22121|2(k\u2212m)\u22121 q . Hence, the number of k-energy tuples with P i\u0131I ai \u2212P j\u2208J bj = 0 is at most \u226a|Sd\u22121|2k\u22122 q2 . 11 \fSumming over all possibilities of sets I and J, we obtain T bad k (Sd\u22121) = o(Tk(Sd\u22121). From the bounds of T dep k , T 0 k , and T bad k (Sd\u22121), we conclude that most of k-energy tuples in Sd\u22121 satisfying (a), (b), and (c). We denote the number of those tuples by T \u2217 k (Sd\u22121). We recall that for any two non-trivial spherical con\ufb01gurations X and X\u2032, they are in the same congruent class if there exists g \u2208O(d, Fq) such that gX = X\u2032. For each con\ufb01guration in Q, say, X = {x0, x0 + x1, x0 + x2, . . . , x0 + x2k\u22122, x0 + 2k\u22122 X i=1 (\u22121)i+1(x + xi)}, the 2k \u22121 vertices x0, x0 + x1, . . . , x0 + x2k\u22122 form a non-degenerate (2k \u22122)-simplex. We know from [3] that the stabilizer of a non-degenerate (2k \u22122)\u2013simplex in Sd\u22121 is of cardinality at least |O(d \u22122k + 1)|. For any X \u2208Q, let \u00b5(X) be the number of con\ufb01gurations which are congruent to X. We have P X\u2208Q \u00b5(X) = T \u2217 k (Sd\u22121). By Cauchy-Schwarz inequality, we have X X\u2208Q \u00b5(X) \u2264|Q|1/2 \u00b7  X X \u00b5(X)2 !1/2 . (3) On the other hand, P X s(X)\u00b5(X)2 is at most the number of pairs of con\ufb01gurations (X, X\u2032) such that X\u2032 = g(X) for some g \u2208(d, Fq), where s(X) is the stabilizer of X. Hence, we can bound P X s(X)\u00b5(X)2 by T \u2217 k (Sd\u22121) \u00b7 |O(d, Fq)|. This implies that X X \u00b5(X)2 \u2264|O(d, Fq)| \u00b7 T \u2217 k (Sd\u22121) |O(d \u22122k + 1)| . (4) We recall from [3] that |O(n, Fq)| \u223cq(n 2). From (3) and (4), we obtain |Q| \u226bq2k2\u22123k. This completes the proof. 4 Proof of Proposition 1.3 Proof of Proposition 1.3. Suppose q = pr with r = 1 \u01eb (assume that 1/\u01eb is an integer). Let A be an arithmetic progression in Fq of size pr\u22121. Let X be the hyperplane xd = 0. De\ufb01ne H := {X + (0, . . . , 0, a): a \u2208A}. 12 \fNote that H is a set of |A| translates of the hyperplane X. We have |H| = qd\u22121 \u00b7 q r\u22121 r = qd\u2212\u01eb. It is not hard to see that |(H \u2212H) \u2229Sd\u22121| \u226aqd\u22122 \u00b7 q r\u22121 r \u226aqd\u22121\u2212\u01eb = o(|Sd\u22121|). For any 1 \u2264m \u226a|Sd\u22121|, let E \u2282Sd\u22121 \\ (H \u2212H) with |E| = m, we have (H \u2212H) \u2229E = \u2205. (5) If \u00b5 < |E| 2q\u01eb , then by Lemma 2.2 for the graph CFd q(E), one has e(H, H) \u2265|H|2|E| qd \u2212|E||H| 2q\u01eb > 0, whenever |H| > qd\u2212\u01eb 2 , which contradicts to (5). In other words, we have \u00b5 \u2265|E| 2q\u01eb . In the case |E + E| = K|E| < qd/2, we start with an observation that T2(E) \u2265 |E|4 |E + E|, which implies T2(E) \u2265|E|4 K|E|. Let X be the multi-set in Fd q de\ufb01ned by X = E + E. We can apply the Expander mixing lemma for the graph CFd q(E) to get an upper bound for T2(E). Indeed, one has T2(E) = e(X, \u2212E) \u2264|E|4 qd + \u00b5 \u00b7 T2(E)1/2 \u00b7 |E|1/2. This gives us T2(E) \u2264|E|4 qd + \u00b52 \u00b7 |E|. Since K|E| < qd/2, we have \u00b52|E| \u226b |E|4 K|E|. This gives \u00b5 \u226b |E| K1/2. Hence, when K \u223c1, we have \u00b5 \u226b|E|. 13 \f5 Proof of Proposition 1.4 Proof of Proposition 1.4. We have seen in the proof of Theorem 1.2 that the number of cycles of length 2k is equal to qd \u00b7 T \u2217 k (Sd\u22121) and T \u2217 k (Sd\u22121) = (1 + o(1))|Sd\u22121|2k\u22121/q. Hence, the number of cycles of length 2k in CFd q(Sd\u22121) is (1 + o(1))|Sd\u22121|2k\u22121qd\u22121. To prove the upper bound on the number of cycles of length 2k in a given set A \u2282Fd q, we need to recall the following result from [4]. Theorem 5.1 (Bennett-Chapman-Covert-Hart-Iosevich-Pakianathan, [4]). For A \u2282Fd q, d \u22652 and an integer k \u22651. Suppose that 2k ln 2q d+1 2 = o(|A|) then the number of paths of length k with vertices in A in CFd q(Sd\u22121) is (1 + o(1))|A|k+1 qk . Let N be the number of cycles of length 2k with vertices in A. For any two vertices x, y \u2208A, let P(x, y) be the number of paths of length k between x and y with vertices in A. It follows from Theorem 5.1 that X x,y\u2208A P(x, y) = (1 + o(1))|A|k+1 qk . It is clear that N = X x,y\u2208A \u0012P(x, y) 2 \u0013 . Using the convexity of the function \u0000x 2 \u0001 , one has N \u226b|A|2 \u00b7 \u0012 P x,y\u2208A P (x,y) |A|2 2 \u0013 \u226b|A|2k q2k , provided that |A| \u226bq k k\u22121 . This completes the proof. Remark 5.1. We remark here that for any k \u22652, there exists a set E \u2286Sd\u22121 with |E| \u226b q d 2k\u22121 such that all cycles of length 2k in CFd q(E) do not have distinct vertices. Such a set can be constructed easily as follows. Let H be a 2k-uniform hypergraph with the vertex set Sd\u22121, and each edge is a good k-energy tuple, then we know from the proof of Proposition 3.5 that the number of edges in H is at most |Sd\u22121|2k\u22121/q. Applying Spencer\u2019s independent hypergraph number lemma in [16], we get an independent set E of size at least \u226bq d 2k\u22121 . This set will satisfy our desired properties. 14 \fAcknowledgments The author was supported by Swiss National Science Foundation grant P4P4P2-191067. I would like to thank Ilya Shkredov for useful discussions about the second eigenvalue of the local Cayley distance graphs."
+                },
+                {
+                    "url": "http://arxiv.org/abs/1905.06483v5",
+                    "title": "Distribution of distances in positive characteristic",
+                    "abstract": "Let $\\mathbb{F}_q$ be an arbitrary finite field, and $\\mathcal{E}$ be a set\nof points in $\\mathbb{F}_q^d$. Let $\\Delta(\\mathcal{E})$ be the set of\ndistances determined by pairs of points in $\\mathcal{E}$. By using the\nKloosterman sums, Iosevich and Rudnev proved that if $|\\mathcal{E}|\\ge\n4q^{\\frac{d+1}{2}}$, then $\\Delta(\\mathcal{E})=\\mathbb{F}_q$. In general, this\nresult is sharp in odd-dimensional spaces over arbitrary finite fields. In this\npaper, we use the recent point-plane incidence bound due to Rudnev to prove\nthat if $\\mathcal{E}$ has Cartesian product structure in vector spaces over\nprime fields, then we can break the exponent $(d+1)/2$, and still cover all\ndistances. We also show that the number of pairs of points in $\\mathcal{E}$ of\nany given distance is close to its expected value.",
+                    "authors": "Thang Pham, Le Anh Vinh",
+                    "published": "2019-05-16",
+                    "updated": "2020-07-30",
+                    "primary_cat": "math.CO",
+                    "cats": [
+                        "math.CO",
+                        "math.NT"
+                    ],
+                    "main_content": "Introduction Let E be a \ufb01nite subset of Rd (d \u22652), and \u2206(E) be the distance set determined by E. The Erd\u02dd os distinct distances problem is to \ufb01nd the best lower bound of the size of the distance set \u2206(E) in terms of the size of the point set E. In the plane case, Erd\u02dd os [7] conjectured that |\u2206(E)| \u226b|E|/ p log |E|. This conjecture was proved up to logarithmic factor by Guth and Katz [10] in 2010. More precisely, they showed that |\u2206(E)| \u226b|E|/ log |E|. In higher dimension cases, Erd\u02dd os [7] also conjectured that |\u2206(E)| \u226b|E|2/d. Interested readers are referred to [27] for results on Erd\u02dd os distinct distances problem in three and higher dimensions. In this paper, we use the following notations: X \u226aY means that there exists some absolute constant C1 > 0 such that X \u2264C1Y , X \u223cY means Y \u226aX \u226aY , X \u2273Y means X \u226b (log Y )\u2212C2Y for some absolute constant C2 > 0, and X \u2273d Y mean X \u2265C3(log Y )\u2212C4Y for some positive constants C3, C4 depending on d. As a continuous analog of the Erd\u02dd os distinct distances problem, Falconer [8] asked how large the Hausdor\ufb00dimension of E \u2282Rd needs to be to ensure that the Lebesgue measure of \u2206(E) is positive. He conjectured that for any subset E \u2282Rd of the Hausdor\ufb00dimension greater than d/2 then E determines a distance set of a positive Lebesgue measure. This conjecture is still open in all dimensions. We refer readers to [6, 9] for recent updates on this conjecture. \u2217Department of Mathematics, ETH Switzerland. Email: phamanhthang.vnu@gmail.com \u2020Vietnam Institute of Educational Sciences. Email: vinhle@vnies.edu.vn 1 \fLet Fq be the \ufb01nite \ufb01eld of order q, where q is an odd prime power. Given two points x = (x1, . . . , xd) and y = (y1, . . . , yd) in Fd q, we denote the distance between x and y by ||x \u2212y|| := (x1 \u2212y1)2 + . . . + (xd \u2212yd)2. Note that the distance function de\ufb01ned here is not a metric but it is invariant under translations and actions of the orthogonal group. For a subset E \u2282Fd q, we denote the set of all distances determined by E by \u2206(E) := {||x \u2212y|| : x, y \u2208E}. The \ufb01nite \ufb01eld analogue of the Erd\u02dd os distinct distances problem was \ufb01rst studied by Bourgain, Katz, and Tao in 2003 [2]. More precisely, they proved that in the prime \ufb01eld Fp with p \u22613 mod 4, for any subset E \u2282F2 p of the cardinality |E| = p\u03b1, 0 < \u03b1 < 2, then |\u2206(E)| \u226b|E| 1 2+\u01eb for some \u01eb = \u01eb(\u03b1) > 0. Note that the condition p \u22613 mod 4 in Bourgain, Katz, and Tao\u2019s result is necessary, since if p \u22611 mod 4, then there exists i \u2208Fp such that i2 = \u22121. By taking E = {(x, ix) : x \u2208Fp}, we have |E| = p and \u2206(E) = {0}. In the setting of arbitrary \ufb01nite \ufb01elds Fq, Iosevich and Rudnev [18] showed that Bourgain, Katz, and Tao\u2019s result does not hold. For example, assume that q = p2, one can take E = F2 p then \u2206(E) = Fp or |\u2206(E)| = |E|1/2. Thus, Iosevich and Rudnev reformulated the problem in the spirit of the Falconer distance conjecture over the Euclidean spaces. More precisely, they asked for a subset E \u2282Fd q, how large does |E| need to be to ensure that \u2206(E) covers the whole \ufb01eld or at least a positive proportion of all elements of the \ufb01eld? Using Fourier analytic methods, Iosevich and Rudnev [18] proved that for any point set E \u2282Fd q with the cardinality |E| \u22654q(d+1)/2 then \u2206(E) = Fq. Hart, Iosevich, Koh, and Rudnev [16] showed that, in general, the exponent (d + 1)/2 cannot be improved when d is odd, even if we only want to cover a positive proportion of all the distances. In even dimensional cases, it has been conjectured that the exponent (d + 1)/2 can be improved to d/2, which is in line with the Falconer distance conjecture in the Euclidean space. In the plane case, Bennett, Hart, Iosevich, Pakianathan, and Rudnev [3] proved that if E \u2282F2 q of cardinality |E| \u2265q4/3, then \u2206(E) covers a positive proportion of all distances. Murphy and Petridis [20] showed that there are in\ufb01nite subsets of F2 q of size q4/3 whose distance sets do not cover the whole \ufb01eld Fq. It is not known whether there exist a small c > 0 and a set E \u2282F2 q with |E| \u2265cq3/2 such that \u2206(E) \u0338= Fq. We refer the interested reader to [16, Theorem 2.7] for a construction in odd dimensional spaces. Chapman et al. [4] broke the exponent d+1 2 to d2 2d\u22121 under the additional assumption that the set E has Cartesian product structure. However, in this case, they can cover only a positive proportion of all distances. In the setting of prime \ufb01elds, it has been proved in [22] that for A \u2282Fp, we have |\u2206(Ad)| \u22651 c \u00b7 min{|A|2\u2212 1 2d\u22122 , p} with c = 2 2d\u22121\u22121 2d\u22122 . Therefore, |\u2206(Ad)| \u2265p c under the condition |A| \u2265p 2d\u22122 2d\u22121\u22121. However, this result again only gives us a 2 \fpositive proportion of all distances, and does not tell us the number of pairs of any given distance. In this paper, we will show that if E \u2282Fd p has Cartesian product structure, we can break the exponent d+1 2 due to Iosevich and Rudnev [18] and still cover all possible distances. Our main tool is the point-plane incidence bound due to Rudnev [24]. Our \ufb01rst result is for odd dimensional cases. Theorem 1.1. Let Fp be a prime \ufb01eld, and A be a set in Fp. For an integer d \u22653, suppose the set A2d+1 \u2282F2d+1 p satis\ufb01es |A2d+1| \u2273d p 2d+2 2 \u22123\u00b72d\u22122\u2212d\u22121 3\u00b72d\u22121\u22121 , then we have \u2022 The distance set covers all elements in Fp, namely, \u2206(A2d+1) = (A \u2212A)2 + \u00b7 \u00b7 \u00b7 + (A \u2212A)2 | {z } 2d+1 terms = Fp. \u2022 In addition, the number of pairs (x, y) \u2208A2d+1 \u00d7 A2d+1 satisfying ||x \u2212y|| = \u03bb is \u223cp\u22121|A|4d+2 for any \u03bb \u2208Fp. Corollary 1.2. For A \u2282Fp, suppose that |A| \u2273p6/11, then we have \u2206(A7) = (A\u2212A)2 + (A\u2212A)2 + (A\u2212A)2 + (A\u2212A)2 + (A\u2212A)2 + (A\u2212A)2 + (A\u2212A)2 = Fp. Our second result is for even dimensional cases. Theorem 1.3. Let Fp be a prime \ufb01eld, and A be a set in Fp. For an integer d \u22653, suppose the set A2d \u2282F2d p satis\ufb01es |A2d| \u2273d p 2d+1 2 \u22122d\u22122d\u22121 2d+1\u22122 , then we have \u2022 The distance set covers all elements in Fp, namely, \u2206(A2d) = (A \u2212A)2 + \u00b7 \u00b7 \u00b7 + (A \u2212A)2 | {z } 2d terms = Fp. \u2022 In addition, the number of pairs (x, y) \u2208A2d\u00d7A2d satisfying ||x\u2212y|| = \u03bb is \u223cp\u22121|A|4d for any \u03bb \u2208Fp. Corollary 1.4. For A \u2282Fp, suppose that |A| \u2273p4/7, then we have \u2206(A6) = (A \u2212A)2 + (A \u2212A)2 + (A \u2212A)2 + (A \u2212A)2 + (A \u2212A)2 + (A \u2212A)2 = Fp. 3 \fRemark 1.1. In the setting of arbitrary \ufb01nite \ufb01elds Fq, we can not break the exponent (d + 1)/2, and still cover all distances with the method in this paper and the distance energy in [21, Lemma 3.1]. More precisely, for A \u2282Fq, one can follow the proofs of Theorems 1.1 and 1.3 to get the conditions |A2d+1| \u226bq 2d+2 2 + 1 4d and |A2d| \u226bq 2d+1 2 + 1 4d\u22122 for odd and even dimensions, respectively. Remark 1.2. The Cauchy-Davenport theorem states that for X, Y \u2282Fp, we have |X + Y | \u2265min{p, |X| + |Y | \u22121}. It is not hard to check that \u2206(A2d) = \u2206(Ad) + \u2206(Ad). The Chapman et al. \u2019s result [4] tells us that |\u2206(Ad)| \u2265p/2 whenever |A| \u226bp d 2d\u22121. Therefore, one can apply the Cauchy-Davenport theorem to show that |\u2206(A2d)| \u2265p \u22121 under the condition |A| \u2265p d 2d\u22121. However, our set A2d lies on the 2d-dimensional space F2d p , thus the exponent d 2d\u22121 is worse than the Iosevich-Rudnev\u2019s exponent 2d+1 4d . The same happens for odd dimensional spaces. Note that the bound |\u2206(Ad)| \u22651 c \u00b7 min{|A|2\u2212 1 2d\u22122 , p} with c = 2 2d\u22121\u22121 2d\u22122 in [22] is not suitable for this approach since the constant factor 1/c is too small. Let Fq be an arbitrary \ufb01nite \ufb01eld, and E \u2282Fd q. The product set of E, denoted by \u03a0(E), is de\ufb01ned as follows: \u03a0(E) := {x \u00b7 y: x, y \u2208E}. Using Fourier analysis, Hart and Iosevich [15] proved that if |E| \u226bq d+1 2 , then \u03a0(E) \u2287 Fq \\ {0}. Moreover, under the same condition on the size of E, we have the number of pairs (x, y) \u2208E \u00d7 E satisfying x \u00b7 y = \u03bb is \u223cq\u22121|E|2 for any \u03bb \u0338= 0. If E has Cartesian product structure, i.e. E = Ad for some A \u2282Fq, then the condition |E| \u226bq d+1 2 is equivalent with |A| \u226bq 1 2+ 1 2d. In the setting of prime \ufb01elds Fp, if d = 8, Glibichuk and Konyagin [12] proved that for A, B \u2282Fp, if |A|\u2308|B|/2\u2309\u2265p, then we have 8A \u00b7 B = Fp. This result has been extended to arbitrary \ufb01nite \ufb01elds by Glibichuk and Rudnev [13]. In this paper, using the techniques in the proof of Theorems 1.3, we are able to obtain the following theorem. Theorem 1.5. For A \u2282Fp, suppose that |A| \u2273p4/7, then we have \u2022 6A \u00b7 A = A \u00b7 A + A \u00b7 A + A \u00b7 A + A \u00b7 A + A \u00b7 A + A \u00b7 A = Fp. \u2022 For any \u03bb \u2208Fp, the number of pairs (x, y) \u2208A6\u00d7A6 such that x\u00b7y = \u03bb is \u223cp\u22121|A|12. Note that our exponent 4/7 improves the exponent 7/12 of Hart and Iosevich [15] in the case d = 6. The following is the conjecture due to Iosevich. Conjecture 1.6. Let A be a set in Fp, suppose that |A| \u226bp 1 2 +\u01eb for any \u01eb > 0, then we have A \u00b7 A + A \u00b7 A = Fp, (A \u2212A)2 + (A \u2212A)2 = Fp. In the spirit of sum-product problems, it has been proved in [22] that for A \u2282Fp, if |A| \u226ap 1 2+ 1 5\u00b72d\u22121\u22122, d \u22652, then we have max \b |\u2206(Ad)|, |\u03a0(Ad)| \t \u226b|A| 2\u2212 1 5\u00b72d\u22123. 4 \fUsing our energies (Lemmas 2.2 and 2.4 below), and the prime \ufb01eld analogue of BalogWooley decomposition energy due to Rudnev, Shkredov, and Stevens [23], we are able to give the energy variant of this result. Theorem 1.7. Let d \u22652 be an integer, A be a set in Fp with |A| \u226ap 1 2+ 1 5\u00b72d\u22121\u22122. There exist two disjoint subsets B and C of A such that A = B \u2294C and max \b Ed((B \u2212B)2), Ed(C \u00b7 C) \t \u226ad4(log |A|)4|A|4d\u22122+ 1 5\u00b72d\u22123 , where Ed ((B \u2212B)2) is the number of 4d-tuples {(ai, bi, ci, ei)}d i=1 with ai, ci, bi, ei \u2208B such that (a1\u2212b1)2+\u00b7 \u00b7 \u00b7+(ad \u2212bd)2 = (c1\u2212e1)2+\u00b7 \u00b7 \u00b7+(cd\u2212ed)2, and Ed (C \u00b7 C) be the number of 4d-tuples {(ai, bi, ci, ei)}d i=1 with ai, ci, bi, ei \u2208C such that a1b1+\u00b7 \u00b7 \u00b7+adbd = c1e1+\u00b7 \u00b7 \u00b7+cded. 2 Preliminaries Let E and F be multi-sets in F2 p. We denote by E and F the sets of distinct elements in E and F, respectively. For any multi-set X, we use the notation |X| to denote the size of X. For \u03bb \u2208Fp, let N(E, F, \u03bb) be the number of pairs ((e1, e2), (f1, f2)) \u2208E \u00d7 F such that e1f1 + e2 + f2 = \u03bb. In the following lemma, we provide an upper bound and a lower bound of N(E, F, \u03bb) for any \u03bb \u2208Fp. Note that, this lemma is essentially the weighted version of the second listed author point-line incidences [28] in the plane F2 q (see also [14, Lemma 14]). Lemma 2.1. Let E, F be multi-sets in F2 p. For any \u03bb \u2208Fp, we have \f \f \f \fN(E, F, \u03bb) \u2212|E||F| p \f \f \f \f \u2264p 1 2 \uf8eb \uf8ed X (e1,e2)\u2208E mE((e1, e2))2 X (f1,f2)\u2208F mF((f1, f2))2 \uf8f6 \uf8f8 1/2 , where mX((a, b)) is the multiplicity of (a, b) in X with X \u2208{E, F}. Proof. Let \u03c7 be a non-trivial additive character on Fp. We have N(E, F, \u03bb) = X (e1,e2)\u2208E,(f1,f2)\u2208F 1 pmE((e1, e2))mF((f1, f2)) X s\u2208Fp \u03c7(s \u00b7 (e1f1 + e2 + f2 \u2212\u03bb)). This gives us N(E, F, \u03bb) = |E||F| p + L, where L = X (e1,e2)\u2208E,(f1,f2)\u2208F mE((e1, e2))mF((f1, f2))1 p X s\u0338=0 \u03c7(s \u00b7 (e1f1 + e2 + f2 \u2212\u03bb)). If we view L as a sum in (e1, e2) \u2208E, then we can apply the Cauchy-Schwarz inequality to 5 \fderive the following: L2 \u2264 X (e1,e2)\u2208E mE((e1, e2))2 X (e1,e2)\u2208F2 p 1 p2 X s,s\u2032\u0338=0 X (f1,f2),(f\u2032 1,f\u2032 2)\u2208F mF((f1, f2))mF((f \u2032 1, f \u2032 2)) \u00b7 \u03c7(s \u00b7 (e1f1 + e2 + f2 \u2212\u03bb))\u03c7(s\u2032 \u00b7 (\u2212e1f \u2032 1 \u2212e2 \u2212f \u2032 2 + \u03bb)) = X (e1,e2)\u2208E mE((e1, e2))2 1 p2 X (e1,e2)\u2208F2 p (f1,f2)\u2208F (f\u2032 1,f\u2032 2)\u2208F s,s\u2032\u0338=0 mF((f1, f2))mF((f \u2032 1, f \u2032 2))\u03c7(e1(sf1 \u2212s\u2032f \u2032 1))\u03c7(e2(s \u2212s\u2032)) \u00b7 \u03c7(s(f2 \u2212\u03bb) \u2212s\u2032(f \u2032 2 \u2212\u03bb)) = X (e1,e2)\u2208E mE((e1, e2))2 X s\u0338=0 (f1,f2)\u2208F (f\u2032 1,f\u2032 2)\u2208F f1=f\u2032 1 mF((f1, f2))mF((f \u2032 1, f \u2032 2))\u03c7(s \u00b7 (f2 \u2212f \u2032 2)) = I + II, where I is the sum over all pairs ((f1, f2), (f1, f \u2032 2)) with f2 = f \u2032 2, and II is the sum over all pairs ((f1, f2), (f \u2032 1, f \u2032 2)) with f2 \u0338= f \u2032 2. It is not hard to check that if f2 \u0338= f \u2032 2, then X s\u0338=0 \u03c7(s \u00b7 (f2 \u2212f \u2032 2)) = \u22121, so II < 0. Note that |II| \u2264I since L2 \u22650. On the other hand, if f2 = f \u2032 2, then X s\u0338=0 \u03c7(s \u00b7 (f2 \u2212f \u2032 2)) = p \u22121. In other words, I \u2264p X (e1,e2)\u2208E mE((e1, e2))2 X (f1,f2)\u2208F mF((f1, f2))2, which implies that |L| \u2264 \u221a I + II \u2264p 1 2 \uf8eb \uf8ed X (e1,e2)\u2208E mE((e1, e2))2 X (f1,f2)\u2208F mF((f1, f2))2 \uf8f6 \uf8f8 1/2 . This completes the proof of the lemma. For A \u2282Fp, let Ed ((A \u2212A)2) be the number of 4d-tuples {(ai, bi, ci, ei)}d i=1 with ai, ci, bi, ei \u2208 6 \fA such that (a1 \u2212b1)2 + \u00b7 \u00b7 \u00b7 + (ad \u2212bd)2 = (c1 \u2212e1)2 + \u00b7 \u00b7 \u00b7 + (cd \u2212ed)2. Similarly, let Ed (A \u00b7 A) be the number of 4d-tuples {(ai, bi, ci, ei)}d i=1 with ai, ci, bi, ei \u2208A such that a1b1 + \u00b7 \u00b7 \u00b7 + adbd = c1e1 + \u00b7 \u00b7 \u00b7 + cded. In our next lemmas, we give recursive formulas for Ed((A \u2212A)2) and Ed(A \u00b7 A). Lemma 2.2. For A \u2282Fp, we have Ed \u0000(A \u2212A)2\u0001 \u2264Cd2(log |A|)2 \u0012|A|4d p + |A|2d+1p Ed\u22121 ((A \u2212A)2) \u0013 , for some positive constant C. The proof of this lemma will be given in the next section. The following result is a direct consequence, which tells us an upper bound of Ed((A \u2212A)2). Corollary 2.3. Let A be a set in Fp. For d \u22652, suppose that |A| \u226b(d log |A|)p1/2, then we have Ed \u0000(A \u2212A)2\u0001 \u226ad2(log |A|)2|A|4d p + d4(log |A|)4|A|4d\u22122+ 1 2d\u22121 . Proof. We prove by induction on d that Ed \u0000(A \u2212A)2\u0001 \u22642C2d2(log |A|)2|A|4d p + 2C2d4(log |A|)4|A|4d\u22122+ 1 2d\u22121 , whenever |A| \u2265 \u221a 2C(d log |A|)p1/2, where the constant C comes from Lemma 2.2. The base case d = 2 follows directly from Lemma 2.2 by using the trivial upper bound |A|3 of E1((A \u2212A)2). Suppose the statement holds for any d\u22121 \u22652, we now prove that it also holds for d. Indeed, by induction hypothesis, we have Ed\u22121 \u0000(A \u2212A)2\u0001 \u22642C2(d \u22121)2(log |A|)2|A|4(d\u22121) p + 2C2(d \u22121)4(log |A|)4|A|4d\u22126+ 1 2d\u22122 (1) \u22642C2d2(log |A|)2|A|4(d\u22121) p + 2C2d4(log |A|)4|A|4d\u22126+ 1 2d\u22122 . On the other hand, it follows from Lemma 2.2 that Ed \u0000(A \u2212A)2\u0001 \u2264Cd2(log |A|)2 \u0012|A|4d p + |A|2d+1p Ed\u22121 ((A \u2212A)2) \u0013 . (2) 7 \fPutting (1) and (2) together, we obtain Ed \u0000(A \u2212A)2\u0001 \u2264 Cd2(log |A|)2|A|4d p + \u221a 2C2d2(log |A|)2|A|2d+1 \u0012 d log |A||A|2(d\u22121) p1/2 + d2(log |A|)2|A|2d\u22123+ 1 2d\u22121 \u0013 . Since \u221a 2C(d log |A|)p1/2 \u2264|A|, we have \u221a 2C2d log |A||A|2(d\u22121) p1/2 \u2264C |A|2d\u22121 p . This implies that Ed \u0000(A \u2212A)2\u0001 \u22642Cd2(log |A|)2|A|4d p + 2C2d4(log |A|)4|A|4d\u22122+ 1 2d\u22121 , completing the proof of the corollary. Similarly, for the case of product sets, we have Lemma 2.4. For A \u2282Fp, we have Ed (A \u00b7 A) \u2264Cd2(log |A|)2 \u0012|A|4d p + |A|2d+1p Ed\u22121 (A \u00b7 A) \u0013 , for some positive constant C. Proof. The proof of this lemma is almost identical with that of Lemma 2.2, so we omit it. Corollary 2.5. Let A be a set in Fp. For d \u22652, suppose that |A| \u226b(d log |A|)p1/2, then we have Ed (A \u00b7 A) \u226ad2(log |A|)2|A|4d p + d4(log |A|)4|A|4d\u22122+ 1 2d\u22121 . Proof. The proof of Corollary 2.5 is identical with that of Corollary 2.3 with Lemma 2.4 in the place of Lemma 2.2, thus we omit it. 2.1 Proof of Lemma 2.2 In the proof of Lemma 2.2, we will use a point-plane incidence bound due to Rudnev [24] and an argument in [26, Theorem 32]. Let us \ufb01rst recall that if R is a set of points in F3 p and S is a set of planes in F3 p, then the number of incidences between R and S, denoted by I(R, S), is the cardinality of the set {(r, s) \u2208R \u00d7 S : r \u2208s}. The following is a version of Rudnev\u2019s point-plane incidence bound, which can be found in [30]. 8 \fTheorem 2.6 (Rudnev, [24, 30]). Let R be a set of points in F3 p and S be a set of planes in F3 p, with |R| \u2264|S|. Suppose that there is no line that contains k points of R and is contained in k planes of S. Then I(R, S) \u226a|R||S| p + |R|1/2|S| + k|S|. Proof of Lemma 2.2: We \ufb01rst have Ed \u0000(A \u2212A)2\u0001 = X t1,t2 r(d\u22121)(A\u2212A)2(t1)r(d\u22121)(A\u2212A)2(t2)f(t1, t2), where r(d\u22121)(A\u2212A)2(t) is the number of 2(d\u22121) tuples (a1, . . . , ad\u22121, b1, . . . , bd\u22121) \u2208A2d\u22122 such that (a1\u2212b1)2+\u00b7 \u00b7 \u00b7+(ad\u22121\u2212bd\u22121)2 = t, and f(t1, t2) is the sum P s r(A\u2212A)2+t1(s)r(A\u2212A)2+t2(s). We now split the sum Ed ((A \u2212A)2) into intervals as follows. Ed \u0000(A \u2212A)2\u0001 \u226a L1 X i=1 L2 X j=1 X t1,t2 f(t1, t2)r(i) (d\u22121)(A\u2212A)2(t1)r(j) (d\u22121)(A\u2212A)2(t2), where L1 \u2264log(|A|2d\u22122), L2 \u2264log(|A|2d\u22122), r(i) (d\u22121)(A\u2212A)2(t1) is the restriction of the function r(d\u22121)(A\u2212A)2(x) on the set Pi := {t: 2i \u2264r(d\u22121)(A\u2212A)2(t) < 2i+1}. Using the pigeon-hole principle two times, there exist sets Pi0 and Pj0 for some i0 and j0 such that Ed \u0000(A \u2212A)2\u0001 \u2264 (2d \u22122)2(log |A|)2 X t1,t2 f(t1, t2)r(i0) (d\u22121)(A\u2212A)2(t1)r(j0) (d\u22121)(A\u2212A)2(t2) \u226a d2(log |A|)22i02j0 X t1,t2 f(t1, t2)Pi0(t1)Pj0(t2). One can check that the sum P t1,t2 f(t1, t2)Pi0(t1)Pj0(t2) is equal to the number of incidences between the point set R of points (\u22122a, e, t1 + a2 \u2212e2) \u2208F3 p with a \u2208A, e \u2208A, t1 \u2208Pi0, and the plane set S of planes in F3 p de\ufb01ned by bX + 2cY + Z = t2 \u2212b2 + c2, where b \u2208A, c \u2208A and t2 \u2208Pj0. Without loss of generality, we can assume that |Pi0| \u2264|Pj0|. To apply Theorem 2.6, we need to bound the maximal number of collinear points in R. The projection of R into the plane of the \ufb01rst two coordinates is the set \u22122A \u00d7 A, thus if a line is not vertical, then it contains at most |A| points from R. If a line is vertical, then it contains at most |Pi0| points from R, but that line is not contained in any plane in S. In 9 \fother words, we can apply Theorem 2.6 with k = |A|, and obtain the following X t1,t2 f(t1, t2)Pi0(t1)Pj0(t2) \u226a|A|4|Pi0||Pj0| p + |A|3|Pi0|1/2|Pj0| + |A|3|Pj0| \u226a|A|4|Pi0||Pj0| p + |A|3|Pi0|1/2|Pj0|. We now fall into the following cases: Case 1: If the \ufb01rst term dominates, we have X t1,t2 f(t1, t2)Pi0(t1)Pj0(t2) \u226a|A|4|Pi0||Pj0| p . Case 2: If the second term dominates, we have X t1,t2 f(t1, t2)Pi0(t1)Pj0(t2) \u226a|A|3|Pi0|1/2|Pj0|. Therefore, Ed \u0000(A \u2212A)2\u0001 \u226ad2(log |A|)22i02j0 \u0012|A|4|Pi0||Pj0| p + |A|3|Pi0|1/2|Pj0| \u0013 \u226ad2(log |A|)2 \u0012|A|4d p + |A|2d+1p Ed\u22121 ((A \u2212A)2) \u0013 . where we have used the facts that \u2022 2i0|Pi0|1/2 \u226a p Ed\u22121 ((A \u2212A)2), \u2022 2j0|Pj0| \u226a|A|2d\u22122, \u2022 2i0|Pi0| \u226a|A|2d\u22122. This completes the proof of the lemma. \u25a1 3 Proof of Theorem 1.1 Proof of Theorem 1.1: Let \u03bb be an arbitrary element in Fp. Let E be the multi-set of points (2x, x2 + (y1 \u2212z1)2 + \u00b7 \u00b7 \u00b7 + (yd \u2212zd)2) \u2208F2 p with x, yi, zi \u2208A, and F be the multi-set of points (\u2212t, t2 + (u1 \u2212v1)2 + \u00b7 \u00b7 \u00b7 + (ud \u2212vd)2) \u2208F2 p with t, ui, vi \u2208A. We have |E| = |F| = |A|2d+1. 10 \fIt follows from Lemma 2.1 that \f \f \f \fN(E, F, \u03bb) \u2212|E||F| p \f \f \f \f \u2264p 1 2 \uf8eb \uf8ed X (e1,e2)\u2208E mE((e1, e2))2 X (f1,f2)\u2208F mF((f1, f2))2 \uf8f6 \uf8f8 1/2 . (3) We observe that if N(E, F, \u03bb) is equal to the number of pairs (x, y) \u2208A2d+1 \u00d7 A2d+1 such that ||x \u2212y|| = \u03bb. From the setting of E and F, it is not hard to see that X (e1,e2)\u2208E mE((e1, e2))2 = |A|Ed((A \u2212A)2), X (f1,f2)\u2208F mF((f1, f2))2 = |A|Ed((A \u2212A)2). (4) Putting (3) and (4) together, we have \f \f \f \fN(E, F, \u03bb) \u2212|A|4d+2 p \f \f \f \f \u2264p 1 2|A|Ed((A \u2212A)2). (5) On the other hand, Corollary 2.3 gives us Ed \u0000(A \u2212A)2\u0001 \u226ad2(log |A|)2|A|4d p + d4(log |A|)4|A|4d\u22122+ 1 2d\u22121 . (6) Substituting (6) into (5), we obtain N(E, F, \u03bb) \u223c|A|4d+2p\u22121 whenever |A2d+1| \u2273d p 2d+2 2 \u22123\u00b72d\u22122\u2212d\u22121 3\u00b72d\u22121\u22121 . Since \u03bb is arbitrary in Fp, the theorem follows. \u25a1 4 Proofs of Theorems 1.3 and 1.5 The proof of Theorem 1.3 is similar to that of Theorem 1.1, but we need a higher dimensional version of Lemma 2.1. Let E and F be multi-sets in F3 p. For \u03bb \u2208Fp, let N(E, F, \u03bb) be the number of pairs ((e1, e2, e3), (f1, f2, f3)) \u2208E \u00d7 F such that e1f1 + e2f2 + e3 + f3 = \u03bb. One can follow step by step the proof of Lemma 2.1 to obtain the following. Lemma 4.1. Let E, F be multi-sets in F3 p. For any \u03bb \u2208Fp, we have \f \f \f \fN(E, F, \u03bb) \u2212|E||F| p \f \f \f \f \u2264p \uf8eb \uf8ed X (e1,e2,e3)\u2208E mE((e1, e2, e3))2 X (f1,f2,f3)\u2208F mF((f1, f2, f3))2 \uf8f6 \uf8f8 1/2 . 11 \fWe are now ready to prove Theorem 1.3. Proof of Theorem 1.3: Let \u03bb be an arbitrary element in Fp. Let E be the multi-set of points (2x1, 2x2, x2 1 + x2 2 + (y1 \u2212z1)2 + \u00b7 \u00b7 \u00b7 + (yd\u22121 \u2212zd\u22121)2) \u2208F3 p with xi, yi, zi \u2208A, and F be the multi-set of points (\u2212t1, \u2212t2, t2 1 + t2 2 + (u1 \u2212v1)2 + \u00b7 \u00b7 \u00b7 + (ud\u22121 \u2212vd\u22121)2) \u2208F3 p with ti, ui, vi \u2208A. We have |E| = |A|2d and |F| = |A|2d. It follows from Lemma 4.1 that \f \f \f \fN(E, F, \u03bb) \u2212|E||F| p \f \f \f \f \u2264p \uf8eb \uf8ed X (e1,e2,e3)\u2208E mE((e1, e2,3 ))2 X (f1,f2,f3)\u2208F mF((f1, f2, f3))2 \uf8f6 \uf8f8 1/2 . (7) We observe that if N(E, F, \u03bb) is equal to the number of pairs (x, y) \u2208A2d \u00d7 A2d such that ||x \u2212y|| = \u03bb. From the setting of E and F, it is not hard to see that X (e1,e2,e3)\u2208E mE((e1, e2, e3))2 = |A|2Ed\u22121((A\u2212A)2), X (f1,f2,f3)\u2208F mF((f1, f2, f3))2 = |A|2Ed\u22121((A\u2212A)2). (8) Putting (7) and (8) together, we have \f \f \f \fN(E, F, \u03bb) \u2212|A|4d p \f \f \f \f \u2264p|A|2Ed\u22121((A \u2212A)2). (9) On the other hand, Corollary 2.3 gives us Ed\u22121 \u0000(A \u2212A)2\u0001 \u226ad2(log |A|)2|A|4d\u22124 p + d4(log |A|)4|A|4d\u22126+ 1 2d\u22122 . (10) Substituting (10) into (9), we obtain N(E, F, \u03bb) \u223c|A|4dp\u22121 whenever |A2d| \u2273d p 2d+1 2 \u22122d\u22122d\u22121 2d+1\u22122 . Since \u03bb is arbitrary in Fp, the theorem follows. \u25a1 Proof of Theorem 1.5: The proof of Theorem 1.5 is similar to that of Theorem 1.3 with Corollary 2.5 in the place of Corollary 2.3. \u25a1 5 Proof of Theorem 1.7 Let us \ufb01rst recall the prime \ufb01eld analogue of Balog-Wooley decomposition energy due to Rudnev, Shkredov, Stevens [23]. 12 \fTheorem 5.1 ([23]). Let A be a set in Fp with |A| \u2264p5/8. There exist two disjoint subsets B and C of A such that A = B \u2294C and max{E+(B), E\u00d7(C)} \u2272|A|14/5, where E+(B) = |{(a, b, c, d) \u2208B4 : a + b = c + d}|, and E\u00d7(C) = |{(a, b, c, d) \u2208C4 : ab = cd}|. We refer the interested reader to [1] for the orginal result over R. The most up to date bound for this result over R is due to Shakan [25]. The following is another corollary of Lemma 2.2. Corollary 5.2. Let A be a set in Fp, and B be a subset of A. For an integer d \u22652, suppose that |A| \u226ap 1 2+ 1 5\u00b72d\u22121\u22122 and E+(B) \u2272|A|14/5, then we have Ed((B \u2212B)2) \u226ad4(log |A|)4|A|4d\u22122+ 1 5\u00b72d\u22123 . Proof. We prove by induction on d that Ed((B \u2212B)2) \u22644C2d4(log |A|)4|A|4d\u22122+ 1 5\u00b72d\u22123 , whenever |A| \u2264(Cp) 1 2 + 1 5\u00b72d\u22121\u22122, where the constant C comes from Lemma 2.2. The base case d = 2 follows directly from Lemma 2.2 and the facts that E1((B \u2212B)2) \u226a E+(B) and |B| \u2264|A|. Suppose the corollary holds for d \u22121 \u22652, we now show that it also holds for the case d. Indeed, it follows from Lemma 2.2 that Ed \u0000(B \u2212B)2\u0001 \u2264Cd2(log |A|)2 \u0012|B|4d p + |B|2d+1p Ed\u22121 ((B \u2212B)2) \u0013 . On the other hand, by induction hypothesis, we have Ed\u22121((B \u2212B)2) \u22644C2(d \u22121)4(log |A|)4|A|4d\u22126+ 1 5\u00b72d\u22124 \u22644C2d4(log |A|)4|A|4d\u22126+ 1 5\u00b72d\u22124 . Thus, using the fact that |B| \u2264|A|, we obtain Ed((B\u2212B)2) \u2264d2(log |A|)2 \u0012 C |A|4d p + 2C2d2(log |A|)2|A|4d\u22122+ 1 5\u00b72d\u22123 \u0013 \u22644C2d4(log |A|)4|A|4d\u22122+ 1 5\u00b72d\u22123 , whenever |A| \u2264(Cp) 1 2 + 1 5\u00b72d\u22121\u22122. Using the same argument, we also have another corollary of Lemma 2.4. Corollary 5.3. Let A be a set in Fp, and C be a subset of A. For an integer d \u22652, suppose that |A| \u226ap 1 2+ 1 5\u00b72d\u22121\u22122 and E\u00d7(C) \u2272|A|14/5, then we have Ed(C \u00b7 C) \u226ad4(log |A|)4|A|4d\u22122+ 1 5\u00b72d\u22123 . 13 \fWe are now ready to prove Theorem 1.7. Proof of Theorem 1.7: It follows from Theorem 5.1 that there exist two disjoint subsets B and C of A such that A = B \u2294C and max{E+(B), E\u00d7(C)} \u2272|A|14/5. One now can apply Corollaries 5.2 and 5.3 to derive max \b Ed((B \u2212B)2), Ed(C \u00b7 C) \t \u226ad4(log |A|)4|A|4d\u22122+ 1 5\u00b72d\u22123 . This completes the proof of the theorem. \u25a1 Acknowledgments: The authors are grateful to the referee for useful comments and corrections. The \ufb01rst listed author was supported by Swiss National Science Foundation grant P400P2-183916. The second listed author was supported by the National Foundation for Science and Technology Development Project. 101.99-2019.318."
+                },
+                {
+                    "url": "http://arxiv.org/abs/1812.11556v1",
+                    "title": "On the structure of distance sets over prime fields",
+                    "abstract": "Let $\\mathbb{F}_q$ be a finite field of order $q$ and $\\mathcal{E}$ be a set\nin $\\mathbb{F}_q^d$. The distance set of $\\mathcal{E}$, denoted by\n$\\Delta(\\mathcal{E})$, is the set of distinct distances determined by the pairs\nof points in $\\mathcal{E}$. Very recently, Iosevich, Koh, and Parshall (2018)\nproved that if $|\\mathcal{E}|\\gg q^{d/2}$, then the quotient set of\n$\\Delta(\\mathcal{E})$ satisfies\n\\[\\left\\vert\\frac{\\Delta(\\mathcal{E})}{\\Delta(\\mathcal{E})}\\right\\vert=\\left\\vert\n\\left\\lbrace\\frac{a}{b}\\colon a, b\\in \\Delta(\\mathcal{E}), b\\ne\n0\\right\\rbrace\\right\\vert\\gg q.\\] In this paper, we break the exponent $d/2$\nwhen $\\mathcal{E}$ is a Cartesian product of sets over a prime field. More\nprecisely, let $p$ be a prime and $A\\subset \\mathbb{F}_p$. If\n$\\mathcal{E}=A^d\\subset \\mathbb{F}_p^d$ and $|\\mathcal{E}|\\gg\np^{\\frac{d}{2}-\\varepsilon}$ for some $\\varepsilon>0$, then we have\n\\[\\left\\vert\\frac{\\Delta(\\mathcal{E})}{\\Delta(\\mathcal{E})}\\right\\vert,\n~\\left\\vert \\Delta(\\mathcal{E})\\cdot \\Delta(\\mathcal{E})\\right\\vert \\gg p.\\]\nSuch improvements are not possible over arbitrary finite fields. These results\ngive us a better understanding about the structure of distance sets and the\nErd\\H{o}s-Falconer distance conjecture over finite fields.",
+                    "authors": "Thang Pham, Andrew Suk",
+                    "published": "2018-12-30",
+                    "updated": "2018-12-30",
+                    "primary_cat": "math.CO",
+                    "cats": [
+                        "math.CO",
+                        "math.NT"
+                    ],
+                    "main_content": "Introduction Let q be an odd prime power, and Fq be the \ufb01nite \ufb01eld of order q. For any two points x = (x1, . . . , xd) and y = (y1, . . . , yd) in Fd q, the distance between them is de\ufb01ned by ||x \u2212y|| = (x1 \u2212y1)2 + \u00b7 \u00b7 \u00b7 + (xd \u2212yd)2. This function is not a norm, but it is invariant under translations, rotations, and re\ufb02ections. Given a set E \u2282Fd q, we de\ufb01ne the distance set \u2206(E) := {||x \u2212y||: x, y \u2208E}. \u2217Department of Mathematics, University of California at San Diego, La Jolla, CA, 92093 USA. Supported by Swiss National Science Foundation grant P2ELP2-175050. Email: v9pham@ucsd.edu \u2020Department of Mathematics, University of California at San Diego, La Jolla, CA, 92093 USA. Supported by an NSF CAREER award and an Alfred Sloan Fellowship. Email: asuk@ucsd.edu 1 \fThe \ufb01nite \ufb01eld variant of the Erd\u02dd os distinct distances problem was \ufb01rst studied by Bourgain, Katz, and Tao in [1], who proved the following theorem. Theorem 1.1 (Bourgain-Katz-Tao, [1]). Suppose q \u22613 mod 4 is a prime. Let E be a set in F2 q. If |E| = q\u03b1 with 0 < \u03b1 < 2, then we have |\u2206(E)| \u226b|E| 1 2+\u03b5, for some positive \u03b5 = \u03b5(\u03b1) > 0. Throughout this paper, we write X \u226bY if there is a positive constant C such that X \u2265CY , and X \u226aY if Y \u226bX. Iosevich and Rudnev [8] observed that the conclusion of Theorem 1.1 can not be extended to arbitrary \ufb01nite \ufb01elds in general. For instance, when q is a square, i.e. q = p2 for some prime p, we can choose E = Fp \u00d7 Fp. One can check that in this case, we have |\u2206(E)| = |E|1/2. Furthermore, if \u22121 is a square number in Fq, i.e. \u22121 = i2 for some i \u2208Fq, then we can choose E = {(t, it) \u2208F2 q : t \u2208Fq}. This set only gives us the distance zero. In light of these constructions, Iosevich and Rudnev [8] made the following reformulation of the distinct distances problem, in the spirit of the Falconer distance conjecture [6].1 Problem 1.2. Let E be a set in Fd q, and \u2206(E) be the set of distinct distances determined by the pairs of points in E. How large does E need to be to guarantee that |\u2206(E)| \u226bq? This problem is now known as the Erd\u02dd os-Falconer distance problem over \ufb01nite \ufb01elds. Using Fourier methods, Iosevich and Rudnev [8] proved that if |E| \u226bq(d+1)/2, then the distance set \u2206(E) covers a positive proportion of all elements in Fq, that is, |\u2206(E)| \u226bq. Hart et al. [7] showed that we can have all distances whenever |E| \u22654q d+1 2 . They also gave constructions for the sharpness of the exponent (d+1)/2 in odd dimensions. However, in even dimensions, it is still possible to break the (d + 1)/2 exponent. Chapman et al. [4] made the \ufb01rst step in this direction by showing that if d = 2, then the exponent 3/2 can be decreased to 4/3, which is directly in line with Wol\ufb00\u2019s result [16] for the Falconer distance problem in R2. It has been conjectured that in even dimensions, the assumption |E| \u226bq d 2 is su\ufb03cient for |\u2206(E)| \u226bq. In a recent work, Iosevich, Koh, and Parshall [9] proved that the exponent d/2 holds for the quotient set of the distance set, which is de\ufb01ned by \u2206(E) \u2206(E) = na b : a, b \u2208\u2206(E), b \u0338= 0 o . The statement of their result is as follows. Theorem 1.3 (Iosevich-Koh-Parshall, [9]). Let Fq be a \ufb01nite \ufb01eld of order q, and E be a set in Fd q. 1The Falconer distance conjecture states that for any compact set E \u2282Rd with the Hausdor\ufb00dimension greater than d/2, the distance set \u2206(E) has positive Lebesgue measure. 2 \f1. If d \u22652 is even and |E| \u22659qd/2, then we have \u2206(E) \u2206(E) = Fq. 2. If d \u22653 is odd and |E| \u22656qd/2, then we have {0} \u222aF+ q \u2282\u2206(E) \u2206(E), where F+ q = {x2 : x \u2208Fq, x \u0338= 0}. Notice that the condition |E| \u226bqd/2 in Theorem 1.3 is sharp over arbitrary \ufb01nite \ufb01elds, even if we wish to cover only a positive proportion of all elements in Fq. Indeed, suppose that q = p2 for some prime p. By setting E = Fd p, we have |E| = q d 2 and |\u2206(E)/\u2206(E)| = |Fp| = q1/2. We refer the interested reader to [9] for more discussions. Let us also remark that it seems di\ufb03cult apply the methods in [9] to the analogous problem of having the product set of the distance set cover a positive proportion of Fq. Using a di\ufb00erent approach, Iosevich and Koh [10] proved that for E \u2282Fd q, if |E| \u226bq d 2 + 1 4, then \u2206(E) \u00b7 \u2206(E) = {a \u00b7 b: a, b \u2208\u2206(E)} = Fq. The main purpose of this paper is to show that if E is a Cartesian product of sets over a prime \ufb01eld Fp, we can break the exponent d/2 and still guarantee that \f \f \f \f \u2206(E) \u2206(E) \f \f \f \f , |\u2206(E) \u00b7 \u2206(E)| \u226bp. Our \ufb01rst two results are for the case of the quotient set, in even and odd dimensions. Theorem 1.4. Let Fp be a prime \ufb01eld, and A \u2282Fp. Then for E = Ad \u2282Fd p with d = 2k, k \u22652 \u2208N, we have \f \f \f \f \u2206(E) \u2206(E) \f \f \f \f = \f \f \f na b : a, b \u2208\u2206(E), b \u0338= 0 o\f \f \f \u2265p 3, whenever |E| \u226bp d 2 \u2212\u03b5 with \u03b5 = d 2 \u00b7 2k\u22122k\u22121\u22121 2k\u22121 . Theorem 1.5. Let Fp be a prime \ufb01eld, and A \u2282Fp. Then for E = Ad \u2282Fd p with d = 2k+1, k \u22652 \u2208N, we have \f \f \f \f \u2206(E) \u2206(E) \f \f \f \f = \f \f \f na b : a, b \u2208\u2206(E), b \u0338= 0 o\f \f \f \u2265p 3, whenever |E| \u226bp d 2 \u2212\u03b5 with \u03b5 = d \u00b7 2k+2\u22122k+1\u22123 2k+3\u22126 . Our next two theorems are for the case of the product set, in even and odd dimensions. 3 \fTheorem 1.6. Let Fp be a prime \ufb01eld, and A \u2282Fp. Then for E = Ad \u2282Fd p with d = 2k, k \u22652 \u2208N, we have |\u2206(E) \u00b7 \u2206(E)| = |{a \u00b7 b: a, b \u2208\u2206(E)}| \u226bp, whenever |E| \u226bp d 2 \u2212\u03b5 with \u03b5 = d 2 \u00b7 2k+1\u22125 5\u00b72k\u22125 . Theorem 1.7. Let Fp be a prime \ufb01eld, and A \u2282Fp. Then for E = Ad \u2282Fd p with d = 2k+1, k \u22652 \u2208N, we have |\u2206(E) \u00b7 \u2206(E)| = |{a \u00b7 b: a, b \u2208\u2206(E)}| \u226bp, whenever |E| \u226bp d 2 \u2212\u03b5 with \u03b5 = d \u00b7 2k+1\u22125 10(2k\u22121). Let us remark that it is not possible to break the exponent d/2 for both quotient set and product set of the distance set over arbitrary \ufb01nite \ufb01elds. For instance, suppose q = p2, and E = Ad \u2282Fq with A = Fp. Then we have |E| = q d 2 and |\u2206(E) \u00b7 \u2206(E)| = | \u2206(E) \u2206(E)| = p = q1/2. 2 Proofs of Theorem 1.4 and Theorem 1.5 To prove Theorems 1.4 and 1.5, we make use of the following results. The \ufb01rst result was given by the \ufb01rst author, Vinh and De Zeeuw [13]. The second was given by Balog [2]. Lemma 2.1. Let Fp be a prime \ufb01eld, and A be a set in Fp. For k \u22652, we have \f \f\u2206(Ak) \f \f \u226bmin n |A|2\u2212 1 2k\u22121 , p o . Lemma 2.2. Let Fq be an arbitrary \ufb01nite \ufb01eld of order q, and B, C be sets in Fq. Suppose that B \u2229C = \u2205and |B||C| \u226bq, then we have \f \f \f \f B \u2212C B \u2212C \f \f \f \f \u2265q 3. Lemma 2.3. Let Fp be a prime \ufb01eld, and A be a set in Fp. For k1, k2 \u22652, we have \u2206(Ak1+k2) = \u2206(Ak1) + \u2206(Ak2). Proof. We \ufb01rst show that \u2206(Ak1+k2) \u2282\u2206(Ak1)+\u2206(Ak2). Let t be an element in \u2206(Ak1+k2). We now prove that t can be presented as a sum of two elements t1 \u2208\u2206(Ak1) and t2 \u2208\u2206(Ak2). Indeed, suppose that t = (x1 \u2212y1)2 + \u00b7 \u00b7 \u00b7 + (xk1 \u2212yk1)2 + (xk1+1 \u2212yk1+1)2 + \u00b7 \u00b7 \u00b7 + (xk1+k2 \u2212yk1+k2)2, where xi, yi \u2208A. Set t1 = (x1 \u2212y1)2 + \u00b7 \u00b7 \u00b7 + (xk1 \u2212yk1)2 and t2 = (xk1+1 \u2212yk1+1)2 + \u00b7 \u00b7 \u00b7 + (xk1+k2 \u2212yk1+k2)2. It is clear that t1 is an element in \u2206(Ak1), t2 is an element in \u2206(Ak2), and t = t1 + t2. This implies that \u2206(Ak1+k2) \u2282\u2206(Ak1) + \u2206(Ak2). We now prove the inverse direction \u2206(Ak1) + \u2206(Ak2) \u2282\u2206(Ak1+k2). 4 \fLet t1 be an element in \u2206(Ak1), t2 be an element in \u2206(Ak2). Suppose that t1 is the distance between x = (x1, . . . , xk1) \u2208Ak1 and y = (y1, . . . , yk1) \u2208Ak1, t2 is the distance between z = (z1, . . . , zk2) \u2208Ak2 and y = (t1, . . . , tk2) \u2208Ak2. Then we have t1 + t2 is the distance between (x1, . . . , xk1, z1 . . . , zk2) \u2208Ak1+k2 and (y1, . . . , yk1, t1, . . . , tk2) \u2208Ak1+k2. Hence, t1 + t2 \u2208\u2206(Ak1+k2). In other words, \u2206(Ak1) + \u2206(Ak2) \u2282\u2206(Ak1+k2). We are ready to prove Theorem 1.4. Proof of Theorem 1.4: Let X be a subset of \u2206(Ak) such that for any x \u2208X we have \u2212x \u0338\u2208X. Without loss of generality, we assume that |X| \u2265|\u2206(Ak)|/2. From Lemma 2.3, we have \u2206(E) = \u2206(Ak) + \u2206(Ak). Hence, \f \f \f \f \u2206(E) \u2206(E) \f \f \f \f \u2265 \f \f \f \f X \u2212(\u2212X) X \u2212(\u2212X) \f \f \f \f . Set B = X and C = \u2212X. It follows from our setting that B \u2229C = \u2205. Therefore, applying Lemma 2.2, we have \f \f \f \f \u2206(E) \u2206(E) \f \f \f \f \u2265p 3, whenever |B||C| \u226bp. Since |B| = |C| = |X| \u226b|\u2206(Ak)|, the condition |B||C| \u226bp is equivalent to |\u2206(Ak)|2 \u226bp. Lemma 2.1 tells us that \f \f\u2206(Ak) \f \f \u226bmin n |A|2\u2212 1 2k\u22121 , p o . Hence, by a direct computation, if |E| \u226bp d 2 \u2212\u03b5 with \u03b5 = d 2 \u00b7 2k\u22122k\u22121\u22121 2k\u22121 , then |A| \u226bp 2k\u22122 2k\u22121. So |\u2206(Ak)|2 \u226bp. This concludes the proof of the theorem. \u25a1 Proof of Theorem 1.5: Let B be a subset of \u2206(Ak) such that |B| \u2265|\u2206(Ak)|/2 and B \u2229\u2212B = \u2205. Let C be a subset of \u2206(Ak+1) such that B \u2282C, C \u2229\u2212C = \u2205, and |C| \u2265 |\u2206(Ak+1)|/2. We note that the condition B \u2282C can be satis\ufb01ed since \u2206(Ak) \u2282\u2206(Ak+1). As in the proof of Theorem 1.4, we have \f \f \f \f \u2206(E) \u2206(E) \f \f \f \f \u2265 \f \f \f \f B \u2212(\u2212C) B \u2212(\u2212C) \f \f \f \f . The condition B \u2229\u2212C = \u2205holds since B \u2282C and C \u2229\u2212C = \u2205. Lemma 2.2 implies that if |B||C| \u226bp, then we have \f \f \f \f \u2206(E) \u2206(E) \f \f \f \f \u2265p 3. Thus, in the rest of the proof, we will clarify the condition |B||C| \u226bp. It follows from our setting that |B||C| \u226b|\u2206(Ak)| \u00b7 |\u2206(Ak+1)|. Applying Lemma 2.1, we get |\u2206(Ak)| \u00b7 |\u2206(Ak+1)| \u226bmin \u001a p2, |A| 2k+2\u22123 2k , p|A|2\u22121 2k , p|A|2\u2212 1 2k\u22121 \u001b . 5 \fIn other words, if |A| \u226bp 2k 2k+2\u22123, i.e. |E| \u226bp d 2 \u2212\u03b5 with \u03b5 = d \u00b7 2k+2\u22122k+1\u22123 2k+3\u22126 , the condition |B||C| \u226bp holds. This completes the proof of the theorem. \u25a1 3 Proofs of Theorem 1.6 and Theorem 1.7 The ideas in the proofs of Theorems 1.6 and 1.7 are similar to those of Theorems 1.4 and 1.5, except that we will use the following lemma in the place of Lemma 2.2. Lemma 3.1 (Proof of Theorem F, [12]). Let Fp be a prime \ufb01eld of order p, and A, B, C, D be sets in Fp. Let N(A, B, C, D) be the number of 8-tuples (a, b, c, d, a\u2032, b\u2032, c\u2032, d\u2032) \u2208(A \u00d7 B \u00d7 C \u00d7 D)2 such that (a \u2212b)(c \u2212d) = (a\u2032 \u2212b\u2032)(c\u2032 \u2212d\u2032). Suppose that |A| = |C|, |B| = |D|, and |A| \u2264|B|, then we have N(A, B, C, D) \u226a|A|2|B|2|C|2|D|2 p + p1/2(|A||B||C||D|)11/8 + |A|11/4|B|4 p1/4 + (|A||C||D|)2. Proof of Theorem 1.6: From Lemma 2.3, we have \u2206(E) = \u2206(Ak) + \u2206(Ak). Thus |\u2206(E) \u00b7 \u2206(E)| = | \u0000\u2206(Ak) + \u2206(Ak) \u0001 \u00b7 \u0000\u2206(Ak) + \u2206(Ak) \u0001 | = |(A \u2212B)(C \u2212D)|, where A = C = \u2206(Ak), B = D = \u2212\u2206(Ak). By the Cauchy-Schwarz inequality, we have |(A \u2212B)(C \u2212D)| \u2265|A|2|B|2|C|2|D|2 N(A, B, C, D) , (1) where N(A, B, C, D) is de\ufb01ned as in Lemma 3.1. Lemma 2.1 gives us that |A| = |B| = |C| = |D| \u226bmin n |A|2\u2212 1 2k\u22121 , p o . Since |E| \u226bp d 2 \u2212\u03b5 with \u03b5 = d 2 \u00b7 2k+1\u22125 5\u00b72k\u22125 , which is equivalent with |A| \u226bp 3\u00b72k\u22121 5\u00b7(2k\u22121), we obtain |A| = |B| = |C| = |D| \u226bp3/5. Under this condition and Lemma 3.1, we achieve N(A, B, C, D) \u226a|A|2|B|2|C|2|D|2 p . (2) Putting (1) and (2) together, the theorem follows. \u25a1 6 \fProof of Theorem 1.7: Since E = A2k+1, we have |\u2206(E) \u00b7 \u2206(E)| \u2265|\u2206(A2k) \u00b7 \u2206(A2k)|. It follows from the proof of Theorem 1.6 that if |A| > p 3\u00b72k\u22121 5(2k\u22121) , then |\u2206(A2k) \u00b7 \u2206(A2k)| \u226bp. Therefore, under the condition |E| \u226bp d 2 \u2212\u03b5 with \u03b5 = (2k + 1) \u00b7 2k+1\u22125 10(2k\u22121) = d \u00b7 2k+1\u22125 10(2k\u22121), we obtain |\u2206(E) \u00b7 \u2206(E)| \u2265|\u2206(A2k) \u00b7 \u2206(A2k)| \u226bp. This completes the proof of the theorem. \u25a1 4 Concluding remarks In the setting of arbitrary \ufb01nite \ufb01elds Fq, Do and Vinh [5] proved that for A \u2282Fq with |A| \u226bq1/2, we have |\u2206(Ak)| \u226bmin \u001a q, |A|2k\u22121 qk\u22121 \u001b . One can follow the proofs of Theorems 1.4 and 1.5 to show that \f \f \f \f \u2206(Ad) \u2206(Ad) \f \f \f \f , \f \f \f \f \u2206(Ad+1) \u2206(Ad+1) \f \f \f \f \u2265q 3, under the condition |A| \u226bq1/2. This matches Theorem 1.3. In the proof of Theorem 1.7, one might try to set A = \u2206(Ak) = C, B = D = \u2212\u2206(Ak+1). This is clear that |A| \u2264|B|. However, in Lemma 3.1, in order to get N(A, B, C, D) \u226a |A|2|B|2|C|2|D|2p\u22121, we need the condition |A| > p3/5. This implies that |A|, |B|, |C|, |D| > p3/5. So we get the same condition on the size of A as in the proof of Theorem 1.6. One might also try to apply the bound |X(Y + Z)| \u226bmin \b (|X||Y ||Z|)1/2, p \t in [15] with X = \u2206(E), Y = Z = \u2206(Ak) or Y = \u2206(Ak), Z = \u2206(Ak+1) to bound |\u2206(E) \u00b7 \u2206(E)|, but the exponents are worse than those of Theorems 1.6 and 1.7. It is not known if Problem 1.2, the Erd\u02dd os-Falconer distance problem over \ufb01nite \ufb01elds, changes over prime \ufb01elds. As we mentioned in the introduction, the exponent (d + 1)/2 can not be improved for odd dimensions over arbitrary \ufb01nite \ufb01elds. The constructions in [7], which demonstrates the sharpness of the exponent (d + 1)/2, were based on the structures of sub\ufb01elds. However, in light of our results, one may be able to break this exponent over prime \ufb01elds."
+                }
+            ],
+            "Sophie Stevens": [
+                {
+                    "url": "http://arxiv.org/abs/2102.05446v1",
+                    "title": "On sum sets of convex functions",
+                    "abstract": "In this paper we prove new bounds for sums of convex or concave functions.\nSpecifically, we prove that for all $A,B \\subseteq \\mathbb R$ finite sets, and\nfor all $f,g$ convex or concave functions, we have\n  $$|A + B|^{38}|f(A) + g(B)|^{38} \\gtrsim |A|^{49}|B|^{49}.$$\n  This result can be used to obtain bounds on a number of two-variable\nexpanders of interest, as well as to the asymmetric sum-product problem. We\nalso adjust our technique to also prove the three-variable expansion result\n  \\[\n  |AB+A|\\gtrsim |A|^{\\frac32 +\\frac3{170}}\\,.\n  \\]\n  Our methods follow a series of recent developments in the sum-product\nliterature, presenting a unified picture. Of particular interest is an\nadaptation of a regularisation technique of Xue, that enables us to find\npositive proportion subsets with certain desirable properties.",
+                    "authors": "Sophie Stevens, Audie Warren",
+                    "published": "2021-02-10",
+                    "updated": "2021-02-10",
+                    "primary_cat": "math.CO",
+                    "cats": [
+                        "math.CO",
+                        "math.NT"
+                    ],
+                    "main_content": "Introduction Given \ufb01nite sets A and B of real numbers, the sum set and product set of A and B are de\ufb01ned as A + B = {a + b : a \u2208A, b \u2208B}, AB = {ab : a \u2208A, b \u2208B}. Erd\u02dd os and Szemer\u00b4 edi conjectured that at least one of |A + A| or |AA| is large with respect to |A|. Speci\ufb01cally, they conjectured the following.1 Conjecture 1 (Erd\u02dd os Szemer\u00b4 edi). For all A \u2286Z a \ufb01nite set, and for all \u01eb > 0, we have |AA| + |A + A| \u226b|A|2\u2212\u01eb. This conjecture remains open, and has given rise to the study of the sum-product phenomenon, which, loosely de\ufb01ned, is the notion that \ufb01nite sets cannot be simultaneously additively and multiplicatively 1In this paper we use the standard notation X \u226aY to mean that there exists an absolute constant c with X \u2264cY . We have Y \u226bX i\ufb00X \u226aY . We write X \u223cY to denote the existence of constants 0 < c1 \u2264c2 so that c1X \u2264Y \u2264c2X. Additionally, the symbols \u2272and \u2273are used to suppress logarithmic factors. 1 \fstructured. Conjecture 1 is believed to be true over the real numbers, where current progress is given by Rudnev and Stevens [12]. There are many variants of this problem in the literature; one family of such variants are concerned with convex functions2. Such results quantify the notion that convex functions destroy additive structure. Some examples of common problems in this area are the following: For A \u2286R a \ufb01nite set, and f be a convex function: \u2022 Is the set A + f(A) always large? \u2022 Is at least one of the sets A + A or f(A) + f(A) is always large? Much research has been done towards these problems and their variants, see for instance [1, 2, 4]. This is also related to the notion of a convex set, that is, a set A = {a1 < a2 < ... < an} such that ai+1 \u2212ai > ai \u2212ai\u22121 for all 2 \u2264i \u2264n \u22121. Any convex set is the image of the interval [n] under some convex function f. Current progress for these problems is given, respectively, by Li and Roche-Newton, [4] and Shkredov [16]. Theorem 1 (Li, Roche-Newton). Let A \u2286R be a \ufb01nite set, and let f be a convex function. Then we have |A + f(A)| \u2273|A| 24 19 . Theorem 2 (Shkredov). Let A \u2286R be a \ufb01nite set, and let f be a convex function. Then we have |A + A| + |f(A) + f(A)| \u2273|A| 100 79 . These problems are also related to expander results. Results of this nature state that some set, de\ufb01ned by (typically polynomial) combinations of elements of A, is always large. Two of the simplest examples of expanders are the sets AA + A = {ab + c : a, b, c \u2208A}, A(A + 1) = {a(b + 1) : a, b \u2208A} which are both expected to have size |A|2\u2212\u01eb for all \u01eb > 0. In fact, the expander A(A + 1) is a special case of the set A + f(A) from above. The current bounds in the literature for these expanders are due to Roche-Newton and Warren [8] and Jones and Roche-Newton [3], respectively. Theorem 3 (Roche-Newton, Warren). For all A \u2286R \ufb01nite, we have |AA + A| \u2273|A| 3 2 + 1 194 . Theorem 4 (Jones, Roche-Newton). For all A \u2286R \ufb01nite, we have |A(A + 1)| \u2273|A| 24 19 . 2In this paper all convex functions considered are strictly convex functions. Furthermore, our results also apply to strictly concave functions. 2 \fMain results The proof of the sum-product result in [12] makes use of a combination of techniques used previously in the real numbers, combined with a technique used to prove sum-product results in \ufb01nite \ufb01elds, see [9]. In this paper we extend these techniques to give both quantitative and qualitative improvements to the problems mentioned above. Note that we make no attempt to optimise the logarithmic factors in our results, since in all cases the polynomial factor exponents are not expected to be tight. Our main result is the following. Theorem 5. Let A, B \u2286R be \ufb01nite sets, and let f and g each be either a convex or concave function. Then we have |A + B|38|f(A) + g(B)|38 \u2273(|A||B|)49. For certain choices of A, B, f, and g, this theorem implies improvements to many of the problems mentioned above. Firstly, we can recover the following improvements to Theorems 1 and 2. Corollary 1. For all A \u2286R \ufb01nite, and f a convex function, we have |A + f(A)| \u2273|A|49/38, |A + A| + |f(A) + f(A)| \u2273|A|49/38. The \ufb01rst inequality follows from setting B = f(A) and g = f \u22121. The second follows from setting B = A and f = g. By slightly adjusting the proof of Theorem 5, we can obtain a better bound for di\ufb00erences. Corollary 2. For all A \u2286R \ufb01nite, and f a convex function, we have |A \u2212A|5|f(A) \u2212f(A)|5 \u2273|A|13. In the case A = [1, . . . , n], this matches the bounds of Schoen and Shkredov [14] and Rudnev and Stevens [12] for estimates on di\ufb00erences and sums of convex sets respectively. Furthermore we match the result of Li and Roche-Newton [4] in the case of few di\ufb00erences, many convex di\ufb00erences. Secondly, we \ufb01nd an asymmetric sum-product result. Corollary 3. For all A, B \u2286R \ufb01nite, we have |AB|38|A + B|38 \u2273(|A||B|)49 This follows from setting A = X, B = Y , f = g = log(x). Corollary 3 appears to be a little studied variant of the asymmetric sum-product problem: One example of a result in this direction is by Solymosi [18], who showed that |A + A||B + B||AB| \u2273|A|2|B|2. There has also been work towards the more di\ufb03cult problem of \ufb01nding a lower bound on |A + B||AC|, see for instance [1], or [5, Theorem 10], where the results are rather of a qualitative nature. The statement of Corollary 3 is particularly interesting in the extremal cases of \u2018few sums\u2019 or \u2018few products\u2019: e.g. if |A| = |B| = N 3 \fand |A + B| \u2272N, then |AB| \u2273N 3 2 + 3 38 . Typically the exponent of 3/2 is a barrier in sum-product estimates, and so in this sense, Corollary 3 is threshold-breaking. Thirdly we give some results demonstrating the principle that \u2018translation destroys multiplicative structure\u2019, in particular improving Theorem 4. Corollary 4. For all A, B \u2286R \ufb01nite, we have |A(A + 1)| \u2273|A|49/38, |AB| + |(A + 1)(B + 1)| \u2273(|A||B|)49/76. Finally, by combining techniques used in the proof of Theorem 5 with the method of Roche-Newton and Warren, we can give an improvement and generalisation of Theorem 3. Theorem 6. Let A, B \u2286R be \ufb01nite sets with |A| \u223c|B|. Then we have |AB + A| \u2273|A| 3 2 + 3 170 . Techniques Here we give an overview of the techniques that we use, hinting at the aspects of our method that are most amenable to future improvements. These techniques can be summarised as follows: 1. The Szemer\u00b4 edi-Trotter theorem gives good bounds on E+ 3 (A, B), especially if we have data of the form rQR(a) \u2265T for each a \u2208A. Similarly, the Szemer\u00b4 edi-Trotter theorem gives good bounds on E+ 3 (f(A), B) for a convex function f, if we have data of the form rQ\u2212R(a) \u2265T for each a \u2208A. 2. Using a regularisation result, we can \ufb01nd a subset C \u2286A so that |C| \u2273|A| and for which we have the additive data rQ\u2212R(a) \u2265T for each c \u2208C. 3. We can count solutions (a, b, c) to a tautological equation of the form a \u2212b = (a + c) \u2212(b + c), where we insist that a\u2212b, a+ c are in certain (di\ufb00erent) sets via third moment energy bounds. This gives an auxiliary energy bound, see Proposition 2 below. 4. A corollary of the regularisation result (see Corollary 5 below) allowing us to upper bound certain products of energies, together with this auxiliary energy bound, leads to the result. Underlying many results about expander sets in R (with few variables) is the Szemer\u00b4 edi-Trotter theorem. It is common knowledge that the Szemer\u00b4 edi-Trotter theorem is particularly strong for \ufb01nding bounds on the third moment energy E+ 3 (A, B), an idea \ufb01rst introduced by Schoen and Shkredov [14]. This, in part, is due to the \u2018trick\u2019 that every element of A can be written as a product of elements of AA and A in at least |A| ways: a = (ab)/b for any choice of b \u2208A (we assume here that 0 / \u2208A). However, if one has additional multiplicative structure on A, say rQR(a) \u2265T for each a \u2208A and some auxiliary sets Q and R and a number T , one can use this information 4 \fin place of the aforementioned \u2018trick\u2019. This gives a third moment energy bound in terms of Q, R and T , the strength of which depends on the strength of the multiplicative information. This is the idea behind the so-called Szemer\u00b4 edi-Trotter sets introduced by Shkredov [16], for which the notation d+(A) (and variants thereof) is used. We note that an analogue of this idea takes place in Fp using the point-line incidence bound of Stevens-de Zeeuw [19] in place of the Szemer\u00b4 edi-Trotter theorem, which naturally produces a bound on the fourth moment energy. For a convex function f, this trick changes as follows: we can obtain bounds on E+ 3 (f(A), B) if we have additive structure on A, say rQ\u2212A(a) \u2265T for all a \u2208A. To bene\ufb01t from the \u2018enhanced energy trick\u2019 described above, we need the appropriate data on Q, R and T . A generic technique for this, \ufb01rst described in [11] and re\ufb01ned in [17], yields a subset C \u2286A with suitable parameters: that is, if E+ 3 (A) \u223c|Dt|t3 for some Dt \u2286A \u2212A, then rDt\u2212A(c) \u2265 |Dt|t|A|\u22121, and |C| \u2265|Dt|t|A|\u22121. A recent expository lemma of Xue [20] enhances the strength of this result, to enable one to take |C| \u2273|A| we use an adaptation of this regularisation result. We conclude this section by considering where improvements to these techniques may be found. Certainly for the real numbers, there is hope that one could \ufb01nd a more optimised subset of A, with the data on Q, R and T optimised for the speci\ufb01c applications within our paper. Indeed, such a \u2018better subset\u2019 is present in the current bounds for the sum-product problem [12]). In [12], an elementary, somewhat geometric, argument justi\ufb01es the existence of the subset used in the context of the sum-product problem. The third item of our list might also be improved as follows: we bound the number of solutions to a \u2212b = (a + c) \u2212(b + c) in terms of the third moment energy. During this argument, we use Cauchy Schwarz to bound a factor of E+ 3/2(A, \u00b7) which appears as a by-product of H\u00a8 older\u2019s inequality. However, it may be possible to directly bound E+ 3/2(A, \u00b7) using other methods. For example, if A is a convex set, then Solymosi and Ruzsa [13] show that E+ 3/2(A, B) \u226a|A + B|3/2 for any set B. In the proof of Theorem 6, we (implicitly) turn to the recent technique of studying the line energy (see e.g. [6, 8]). We would not be surprised if future developments of this concept provide further tools relevant to the results in this paper. 1 Preliminaries We use the notation rQ\u2212R(a) to denote the number of representations of the element a as a product from Q \u2212R, that is, rQ\u2212R(a) = |{(q, r) \u2208Q \u00d7 r : q \u2212r = a}|, and similarly for rQR(a) etc. The kth moment additive energy between sets A and B is de\ufb01ned to be E+ k (A, B) := X x\u2208A\u2212B rk A\u2212B(x) for k \u22651. If A = B we simply write E+ k (A). Similarly, we de\ufb01ne the multiplicative energy E\u00d7 k (A, B) := P x rk A/B(x). 5 \f1.1 Energy Bounds via Szemer\u00b4 edi Trotter Before beginning the proofs, we require some technical lemmas. The \ufb01rst gives a bound for the additive energy of two sets A and B, subject to multiplicative information on the set A, and can be found in [12]. We give the proof for completeness, noting that the proof for Lemma 2 follows from a similar argument. Lemma 1. Let A, B, C, Q, R \u2282R be \ufb01nite sets with the property that rQR(a) \u2265T for all a \u2208A and some T \u22651. Then if |R||C| \u226a(|Q||B|)2 , |{c = a \u2212b; a \u2208A, b \u2208B, c \u2208C}| \u226a(|Q||R||B||C|)2/3 T . (1) Furthermore, if |R||A| \u2264|Q|2|B|, E3(A, B) \u226a|Q|2|R|2|B|2 T 3 log |A| . (2) The next lemma bounds the additive energy of two sets f(A) and B, where f is a convex function, subject to additive information on the set A. Lemma 2. Let A \u2282R be \ufb01nite, and let f be a convex (or concave) function. Suppose that there exist \ufb01nite sets Q, R \u2286R with |Q| \u2265|R| and a number T \u22651 so that rQ\u2212R(a) \u2265T for all a \u2208A. Then for any set B satisfying |R||A| \u226a|Q|2|B|, we have E+ 3 (f(A), B) \u226a|Q|2|R|2|B|2 T 3 log |A| . We remark that we have stated Lemmas 1 and 2 as a third energy bound. The same technique with an additional interpolation argument gives us kth moment energy bounds, see e.g. [12] for details. Proof. To prove the \ufb01rst bound, we note that by utilising the information on the sets Q and R, we have |{c = a \u2212b : a \u2208A, b \u2208B, c \u2208C}| \u22641 T |{c = qr \u2212b : q \u2208Q, r \u2208R, b \u2208B, c \u2208C}|, which can be viewed as incidences between the set of lines L given by y = qx \u2212c for (q, c) \u2208Q \u00d7 C, and the point set P = R \u00d7 B. Applying the Szemer\u00b4 edi Trotter theorem, we have |{c = qr \u2212b : q \u2208Q, r \u2208R, b \u2208B, c \u2208C}| = I(P, L) \u226a(|Q||R||B||C|)2/3 + |Q||B|. Because of the constraint present in the statement of the lemma, the leading term dominates. We therefore have |{c = a \u2212b : a \u2208A, b \u2208B, c \u2208C}| \u226a(|Q||R||B||C|)2/3 T as needed. 6 \fFor the second part of the lemma, we decompose the support of E+ 3 (A, B) into dyadic groups: for i = 0, . . . \u230alog |A|\u230b, let Di := {d \u2208A \u2212B : rA\u2212B(d) \u2208[2i, 2i)} \u2286A \u2212B. Then E+ 3 (A, B) = \u230a|A|\u230b X i=0 X d\u2208Di r3 A\u2212B(d) < X i |Di|23i+3 \u226alog |A| max i |Di|23i . With Di playing the role of C in (1), we have 2i|Di| \u2264|{(a, b, d) \u2208A \u00d7 B \u00d7 Di : d = a \u2212b}| \u226a(|Q||R||B||Di|)2/3 T . The result then follows, and all that is left to do is to verify the condition required, for C = Di, i.e. that |Q||Di| \u226a(|R||B|)2. Note that since Di \u2286A \u2212B, this is certainly true if we have |Q||A||B| \u226a(|R||B|)2 \u21d0 \u21d2|Q||A| \u226a|R|2|B| which is the stated condition. 1.2 Regularisation Results In this section we give some regularisation results required for the proof. The \ufb01rst is a lemma present in [12]. This lemma will be used to give a certain subset of A on which much of the energy is supported, and with certain popularity properties. Lemma 3. Let R\u01eb be a deterministic rule with parameter \u01eb \u2208(0, 1) that, to every su\ufb03ciently large \ufb01nite additive set X, associates a subset R\u01eb(X) \u2286X of cardinality |R\u01eb(X)| \u2265(1 \u2212\u01eb)|X|. For any such rule R\u01eb, any m > 1 and a su\ufb03ciently large \ufb01nite set A, set \u01eb = c1 log\u22121(|A|) for some c1 \u2208(0, 1). Then there exists a set B \u2286A (depending on R\u01eb, m), with |B| \u2265(1 \u2212c1)|A| such that E+ m(R\u01eb(B)) \u2265c2 E+ m(B) , for some constant c2 = c2(m, c1) in (0, 1]. We also require the following proposition. It is very similar to an expository lemma of Xue [20, Lemma 5.1], but has been amended to admit an asymmetric form. We present the rather technical proof of this proposition in the appendix, where we make the dependence on log(|A|) and k hidden in the notation explicit. Proposition 1. Let A, V be \ufb01nite subsets of R, let k > 1 be a real number and \ufb01x c1 \u2208(0, 1). Then there are sets B, C with C \u2286B \u2286A and |C| \u2273k,c1 |B| \u2265(1 \u2212c1)|A| such that the following property holds: there is a number 1 \u2264t \u2264|B| and a set Dt = {x \u2208B \u2212V : t \u2264rB\u2212V (x) < 2t} such that E+ k (B, V ) \u223ck |Dt|tk and rDt+V (c) \u223ck |Dt|t |V | for any c \u2208C. 7 \fOn a high level, the proofs of Lemma 3 and Proposition 1 follow the same schemata: given a set A, we de\ufb01ne a rule which extracts a positive proportion subset A\u2032 \u2286A with desirable properties according to the rule in question. In Lemma 3, this rule is abstract, whereas in Proposition 1 it is explicit. We then iterate this procedure until some stopping condition is satis\ufb01ed. In Lemma 3, this stopping condition is relative to the mth energy; in Proposition 1, the stopping condition is de\ufb01ned with respect to the support of the kth energy. These two regularisation results di\ufb00er primarily because of this subtlety. Finally, we argue that this procedure must terminate in an acceptable number of steps, thus eventually outputting a positive proportion subset B \u2286A. Proposition 1 admits the following corollary, which is similar to a result of Shakan [15, Theorem 1.10]. Corollary 5. Let A, V \u2286R be \ufb01nite, and f be a convex (or concave) function. Then there are sets B, C with C \u2286B \u2286A and |C| \u2273k |B| \u226bk |A| such that E+ 3 (B, V )E+ 3 (f(C), U) \u2272|U|2|V |2|A|3 for any set U with |U||V | \u226b|A|. Proof. We apply Proposition 1 with k = 3 to obtain the sets B and C, so that E+ 3 (B, V ) \u223c|Dt|tk where rB\u2212V (d) \u2208[t, 2t) for all d \u2208Dt and rDt+V (c) \u226b|Dt|t|B|\u22121 for all c \u2208C. We are able to obtain a bound on E+ 3 (f(C), U) using Lemma 2: E+ 3 (f(C), U) \u2272|Dt|2|V |2|U|2 |Dt|3t3|B|\u22123 \u223c|V |2|U|2|A|3 E+ 3 (B, V ) . We remark that since |U||V | \u2273|A|, it follows that min(|Dt|, |V |)|C| \u2272max(|Dt|, |V |)2|U| and so we may indeed apply Lemma 2. 2 Auxiliary energy bounds A unifying idea behind the proofs in this paper is the following proposition: Proposition 2. Let A, C \u2286R be \ufb01nite, and k \u22651. Suppose that E+ k (A) \u223c|D|\u2206k for some D \u2286A \u2212A and \u2206\u22651, where rA\u2212A(d) \u2208[\u2206, 2\u2206). Then we have |D|9\u220612 \u2272|A + C|6E+ 3 (A)4E+ 3 (C)2E+ 3 (A, D)E+ 3 (C, A + C)2 |C|18|A|3 . (3) The stated form of Proposition 2 gives us a great deal of \ufb02exibility. For example, if we had multiplicative information on the set A in the guise of Lemma 2 \u2013 that is, if rQR(a) \u2265T for all a \u2208A \u2013 then we obtain an energy estimate in terms of this data. Proposition 2 also admits a multiplicative form, in which E\u00d7 k (A) \u223c|D|\u2206k. Then all instances of E3 in (3) should be replaced by E\u00d7 3 , and A+C by AC. 8 \fProof. We begin by de\ufb01ning the popular set P(A, C) := \u001a x \u2208A + C : rA+C(x) \u2265 |A||C| log |A||A + C| \u001b . We also de\ufb01ne the set A\u2032 := \u001a a \u2208A : |{c \u2208C : a + c \u2208P(A, C)}| \u2265|C| 2 \u001b . We perform a re\ufb01nement step at the beginning of the proof, making use of Lemma 3. We claim that Lemma 3 can be applied with the deterministic rule being the subset A\u2032 \u2286A de\ufb01ned above. Firstly we prove that |A\u2032| is large with respect to |A|. We have X a\u2208A\u2032 |{c \u2208C : a + c \u2208P(A, C)}| + X a\u2208A\\A\u2032 |{c \u2208C : a + c \u2208P(A, C)}| = |{(a, c) \u2208A \u00d7 C : a + c \u2208P(A, C)}| \u2265 \u0012 1 \u2212 1 log |A| \u0013 |A||C|. By setting |A \\ A\u2032| = c|A| and using the bounds X a\u2208A\u2032 |{c \u2208C : a + c \u2208P(A, C)}| \u2264(1 \u2212c)|A||C| X a\u2208A\\A\u2032 |{c \u2208C : a + c \u2208P(A, C)}| \u2264c 2|A||C| we conclude that |A\u2032| \u2265 \u0010 1 \u2212 2 log |A| \u0011 |C|. We can therefore apply Lemma 3 at the outset of the proof, obtaining a set A\u2032 as above with the property that |A\u2032| \u2273|A|, and E+ k (A\u2032) \u223cE+ k (A). We now consider the number of solutions (a, b, c) \u2208A2 \u00d7 C to the trivial equation a \u2212b = (a + c) \u2212(b + c) (4) where the di\ufb00erence a \u2212b comes from the set D \u2286A\u2032 \u2212A\u2032 such that |D|\u2206k \u223cE+ k (A\u2032) \u223cE+ k (A), and such that the sum a + c is popular, that is, a + c \u2208P(A, C). There are at least \u2126(|C||D|\u2206) solutions to equation (4). We partition solutions to (4) with the relevant conditions, via the following: (a, b, c) \u223c(a + t, b + t, c \u2212t), t \u2208R , and let [a, b, c] represent this equivalence class. Since t cancels out in equation (4), these classes are non-trivial. Let N denote the number of solutions to equation (4). We have |C||D|\u2206\u226aN = X [a,b,c] |[a, b, c]| 9 \fand so, after an application of the Cauchy-Schwarz inequality, we obtain (|C||D|\u2206)2 \u2264|{equivalence classes}| \u00b7 X [a,b,c] |[a, b, c]|2. (5) We now aim to bound the two factors in equation (5). To bound the number of equivalence classes, note that each equivalence class gives a solution to the equation d = s1 \u2212s2, d \u2208D, s1 \u2208P(A, C), s2 \u2208A + B. Therefore we have |{equivalence classes}| \u2264|{(d, s1, s2) \u2208D \u00d7 P(A, C) \u00d7 A + C : d = s1 \u2212s2}|. By the popularity of s1, we have |{equivalence classes}| \u2272|A + C| |A||C| |{(d, a, c, s) \u2208D \u00d7 A \u00d7 C \u00d7 A + C : d \u2212a = c \u2212s}| = |A + C| |A||C| X x rD\u2212A(x)rC\u2212(A+C)(x) \u2264|A + C| |A||C| E+ 3 (A, D) 1 6 (|A||D|) 1 2 E+ 3 (C, A + C) 1 3 , where the \ufb01nal bound is an application of H\u00a8 older\u2019s inequality, followed by Cauchy-Schwarz. We now aim to bound the sum X [a,b,c] |[a, b, c]|2 where is it understood that the sum is taken over equivalence classes satisfying the relevant conditions. Note that this sum counts pairs of triples from the same equivalence class, and for each pair we have (a, b, c) \u223c(a\u2032, b\u2032, c\u2032) = \u21d2\u2203t with a \u2212a\u2032 = b \u2212b\u2032 = c\u2032 \u2212c = t. We therefore have X [a,b,c] |[a, b, c]|2 \u2264 X t rA\u2212A(t)2rC\u2212C(t) \u2264E+ 3 (A) 2 3 E+ 3 (C) 1 3 where again, the \ufb01nal inequality is a result of H\u00a8 older\u2019s estimate. Finally, from (5) we have (|C||D|\u2206)2 \u2272|A + C| |A||C| E+ 3 (A, D) 1 6 (|A||D|) 1 2 E+ 3 (C, A + C) 1 3 E+ 3 (A) 2 3 E+ 3 (C) 1 3 . Rearranging and raising both sides to the sixth power concludes the proof. 10 \f3 Proof of Theorem 5 We actually prove the following, slightly more general theorem. Theorem 7. Let f and g be convex or concave functions. Let A, B \u2286R be \ufb01nite sets. Then we have |A|49|B|49 \u2272|A + B|38|f(A) + g(B)|38 (6) and |A|39|B|39 \u2272|A \u00b1 B|20|f(A) \u00b1 g(B)|20|A \u2212A|5||B \u2212B|5|f(A) \u2212f(A)|5|g(B) \u2212g(B)|5 . (7) We clarify that in this theorem, one may take f to be convex, and g to be concave. On a high level, the proof proceeds by apply two iterations of Corollary 5 to A and B with judicious choices of V in each case. Then we apply Proposition 2 to the ensuing subsets and their convex (resp. concave) counterparts. This gives an additive energy relation. We obtain the statements of Theorem 7 using Cauchy-Schwarz and H\u00a8 older inequalities. Let us make two simple observations regarding the third moment energy of a set X that we use in the subsequent argument.Firstly, note that E+ 3 (\u2212X, X \u2212X) = E+ 3 (X, X \u2212X) due to the symmetry of the di\ufb00erence set. Secondly, for any set Y \u2286X and any set Z, we have E+ 3 (Y, Z) \u2264E+ 3 (X, Z). Proof. Here we prove the slightly more technical statement (6), and indicate the changes necessary to prove (7). We begin by applying Corollary 5 to the set A with V = A to obtain sets A2 \u2286A1 \u2286A with |A2| \u2273|A1| \u226b|A| and E3(A1, A)E3(f(A2), U) \u2272|A|5|U|2 for any U . Note that if |U| \u226b1, then this follows from Corollary 5; if |U| \u226a1, then it follows trivially. We now apply the concave analogue of Corollary 5, this time to the set f(A2) with V = f(A) and the function f \u22121. We obtain 3 the sets A4 \u2286A3 \u2286A2 with |A4| \u2273|A3| \u226b|A2| \u2273|A| so that E3(f(A3), f(A))E3(A4, U) \u2272|A|5|U|2 for any U . We repeat this argument for the set B taking V = B to obtain B2 \u2286B1 \u2286B so that E3(B1, B)E3(g(B2), U) \u2272|B|5|U|2 for any U , and then once more to g(B2) with V = g(B) and function g\u22121 to obtain B4 \u2286B3 \u2286B2 with |B4| \u2273|B| and E3(g(B3), g(B))E3(B4, U) \u2272|B|5|U|2 for any U . 3Strictly speaking, we obtain sets f(A4) \u2286f(A3) \u2286f(A2). 11 \fTo prove (6), we dyadically decompose the sets A4, B4, f(A4), g(B4) according to the second moment energy to obtain sets Di and numbers ti \u22651 so that E+ 2 (A4) \u223c|D1|t2 1, E+ 2 (B4) \u223c|D2|t2 2 E+ 2 (f(A4)) \u223c|D3|t2 3, E+ 2 (g(B4)) \u223c|D4|t2 4 . To prove (7), we would instead dyadically decompose the sets A4, B4, f(A4), g(B4) according to the 12/7th moment energy, so that e.g. E+ 12/7(A4) \u223c|D1|t12/7 1 . Note that e.g. D1 \u2286A4 \u2212A4. We now apply Proposition 2 to each of the sets A4, B4, f(A4), g(B4), choosing C in (3) to be B4, A4, g(B4), f(A4) respectively. We then multiply together the four instances of (3), and make liberal use of the simple observations noted at the beginning of this section together with the consequences of Corollary 5. This gives Y 1\u2264i\u22644 |Di|7t12 i \u2272|A + B|20|f(A) + g(B)|20|A|9|B|9 . (8) To prove statement (7), we recall that we had initially dyadically decomposed according to the 12/7th energy and so, after an application of H\u00a8 older\u2019s inequality for E+ 12/7(A4) etc., we are done. To prove statement (6), let us multiply (8) on both sides by (t1t2t3t4)2. Note that |D1|t3 1|D3|t3 3 \u2264E+ 3 (A4, A)E+ 3 (f(A3), f(A)) \u2272|A|7 = \u21d2t1t2 \u2272 |A|7 E+ 2 (A)E+ 2 (f(A)) and similarly (t2t4) \u2272 |B|7 E+ 2 (B)E+ 2 (g(B)) . Hence we obtain \u0000E+ 2 (B)E+ 2 (g(B))E+ 2 (A)E+ 2 (f(A)) \u00019 \u2272|A + B|20|f(A) + g(B)|20|A|23|B|23 . Finally, we use the Cauchy-Schwarz relation |X|2|Y |2 |X + Y | \u2264E+ 2 (X, Y ) \u2264E+ 2 (X)1/2E+ 2 (Y )1/2 to complete the proof. 4 Proof of Theorem 6 In this section we prove Theorem 6 proving two complementary bounds, using a combination of the methods found in [9], [12], and [8]. 12 \f4.1 Proof of Theorem 6 Bound 1 The method of Roche-Newton and Warren [8] involved studying the line energy of lines of a particular structure. Their results, combined with an incidence theorem of Rudnev and Shkredov [10] and an additive combinatorial result of Roche-Newton and Rudnev4 [7] imply the following incidence bound. See also [6] for more information on line energy and its applications. Theorem 8. Let L be a set of lines of the form y = ax + a\u2032 for a, a\u2032 \u2208A \u2286R \\ {0} a \ufb01nite set. Let B, C \u2286R be two \ufb01nite sets. Then we have I(B \u00d7 C, L) \u2272E\u00d7 4 (A) 1 12 |A| 7 6 |B| 2 3 |C| 1 2 + |A|2|C| 1 2 . We shall apply Theorem 8 to the point set B \u00d7 (AB + A) and to the set of lines L of the form y = ax + a\u2032 with a, a\u2032 \u2208A. Note that without loss of generality we may remove 0 from A if it is present. For each line y = ax + a\u2032, for each b \u2208B the point (b, ab + a\u2032) lies on this line, and so we have at least |A|2|B| \u223c|A|3 incidences. Using Theorem 8 we obtain |A|3 \u2272I(B \u00d7 (AB + A), L) \u2272E\u00d7 4 (A) 1 12 |A| 7 6 |B| 2 3 |AB + A| 1 2 + |A|2|AB + A| 1 2 . (9) Note that if the second term dominates we have a much stronger result than claimed in the statement in Theorem 6. Let us therefore assume the \ufb01rst term dominates. Hence we have the \ufb01rst of our two bounds: |A|7/3 \u2272|AB + A|E\u00d7 4 (A)1/6 \u2264|AB + A||A| 1 6 E\u00d7 3 (A)1/6. (10) 4.2 Proof of Theorem 6 Bound 2 To \ufb01nd the second bound, let us apply the multiplicative version of Proposition 1 to the set A with V = A to obtain sets A2 \u2286A1 \u2286A with |A2| \u2273|A1| \u226b|A| so that E\u00d7 3 (A1, A)E+ 3 (A2, U) \u2272|A|5|U|2 . (11) Equation (11) is a consequence of using Lemma 1 in place of Lemma 2 in the proof of Corollary 5. We now apply Proposition 2 to the set A2, writing E+(A2) \u223c|D|t2 and taking C = \u03bbA2 for \u03bb \u0338= 0. Note that E+ k (A2, X) = E+ k (\u03bbA2, \u03bb\u22121X) for any set X and any k \u22651. From Proposition 2, (11), and the inclusions A2 \u2286A1 \u2286A we obtain |D|7t12 \u2272|A + \u03bbA|10 |A|9 \u0012E+ 3 (A2) |A|2 \u001312 E+ 3 (A2, D) |D|2 \u0012E+ 3 (A2, \u03bb\u22121A + A |A + \u03bbA|2 \u00132 \u2272|A + \u03bbA|10 |A|9 |A|45 E\u00d7 3 (A1, A)9 . (12) 4The result of Roche-Newton and Rudnev is that the number of solutions to the equation a1 \u2212a2 a3 \u2212a4 = a5 \u2212a6 a7 \u2212a8 with each ai \u2208A is at most O(|A|6 log |A|). 13 \fWe have |D|t3 \u2264E+ 3 (A2) \u2272 |A|7 E\u00d7 3 (A1, A) = \u21d2t \u2272 |A|7 E\u00d7 3 (A1, A)E+ 2 (A2) , and so multiplying (12) by t2 and applying the Cauchy-Schwarz energy bound |A2|4 |A2 + \u03bbA2| \u2264E(A2, \u03bbA2) \u2264E(A2) we conclude that |A + \u03bbA|19 \u2273E\u00d7 3 (A1, A)11 |A|14 . (13) 4.3 Proof of Theorem 6 Conclusion Combining the bounds of the previous section we obtain |A + BA| \u2273max   E\u00d7 3 (A, A1) 11 19 |A| 14 19 , |A| 13 6 E\u00d7 3 (A, A1) 1 6 ! where the \ufb01rst bound has instead been applied to the set A1 given above, making use of the inequalities |A| \u223c|A1| and E+ 3 (A1) \u2264E+ 3 (A, A1). In the worst possible case, both maximands are equal. This happens if E+ 3 (A1, A) = |A| 331 85 and so we shall assume that this is indeed the case. We then obtain |A + BA| \u2273|A| 129 85 = |A| 3 2 + 3 170 as required. Acknowledgements The authors were supported by Austrian Science Fund FWF Project P 30405-N32. We are especially grateful to Oliver Roche-Newton who pushed us to improve our results. We also thank Misha Rudnev for helpful suggestions."
+                }
+            ],
+            "Alina Ostafe": [
+                {
+                    "url": "http://arxiv.org/abs/1109.0575v1",
+                    "title": "Degree Growth, Linear Independence and Periods of a Class of Rational Dynamical Systems",
+                    "abstract": "We introduce and study algebraic dynamical systems generated by triangular\nsystems of rational functions. We obtain several results about the degree\ngrowth and linear independence of iterates as well as about possible lengths of\ntrajectories generated by such dynamical systems over finite fields. Some of\nthese results are generalisations of those known in the polynomial case, some\nare new even in this case.",
+                    "authors": "Alina Ostafe, Igor Shparlinski",
+                    "published": "2011-09-02",
+                    "updated": "2011-09-02",
+                    "primary_cat": "math.NT",
+                    "cats": [
+                        "math.NT",
+                        "math.DS",
+                        "11T06, 37P05, 37P25, 65C10"
+                    ],
+                    "main_content": "Introduction Let F be an arbitrary \ufb01eld F and let F1, . . . , Fm \u2208F(X1, . . . , Xm) be m rational functions in m variables over F. For each i = 1, . . . , m we de\ufb01ne the 1 \fk-th iteration of the rational function Fi by the recurrence relation F (0) i = Xi, F (k) i = Fi \u0010 F (k\u22121) 1 , . . . , F (k\u22121) m \u0011 , k = 1, 2, . . . . (1) In this paper we consider dynamical systems generated by multivariate rational functions, we refer to [1, 21, 22] for a background on algebraic dynamical systems. More precisely, we de\ufb01ne the vectors un = (un,1, . . . , un,m) \u2208Fm by the recurrence relation un+1,i = Fi(un,1, . . . , un,m), n = 0, 1, . . . , i = 1, . . . , m, (2) with some initial vector u0 = (u0,1, . . . , u0,m) \u2208Fm. As we work with rational functions, we make the standard convention (see [3, 9, 10]) that 0\u22121 = 0. (3) Using the following vector notation F = (F1(X1, . . . , Xm), . . . , Fm(X1, . . . , Xm)), we have the recurrence relation un+1 = F(un), n = 0, 1, . . . . (4) In particular, for any n \u22650 and i = 1, . . . , m we have un,i = F (n) i (u0) = F (n) i (u0,1, . . . , u0,m) or un = F(n)(u0), provided that un has been generated by (4) without using the convention (3) (that is, no poles have been encountered). Clearly, if we work over a \ufb01nite \ufb01eld of q elements, the above sequence (4) of vectors {un} is eventually periodic with some period \u03c4 \u2264qm. One of the important characteristics of the dynamical system generated by F1, . . . , Fm \u2208F(X1, . . . , Xm) is the degree growth of the functions (1). It is of great interest for the theory of dynamical systems and has been studied in a number of works, see, for example, [2, 25] and references therein. It is also important for applications to pseudorandom number generators [24]. 2 \fMore precisely, although for a \u201ctypical\u201d system an exponential degree growth is expected, there are several examples of systems where the degree grows much slower (which is highly bene\ufb01cial for their applications), and such systems are of special interest. For example, in [14, 17] several types of multivariate polynomial systems F = {F1, . . . , Fm} of m polynomials in m variables over a \ufb01nite \ufb01eld Fq have been constructed and studied, having the \u201ctriangular\u201d form F1(X1, . . . , Xm) = X1G1(X2, . . . , Xm) + H1(X2, . . . , Xm), . . . Fm\u22121(X1, . . . , Xm) = Xm\u22121Gm\u22121(Xm) + Hm\u22121(Xm), Fm(X1, . . . , Xm) = gmXm + hm, (5) with Gi, Hi \u2208Fq[Xi+1, . . . , Xm], i = 1, . . . , m \u22121, and gm, hm \u2208Fq, gm \u0338= 0. These systems have been further investigated in [13, 18, 19, 20]. For the systems (5), in the case of constant polynomials Gi \u2208F\u2217 q in [14] and polynomials Gi with leading terms of special form in [17, 18], a series of results have been obtained about the distribution of the corresponding sequences given by (2) that are much stronger than those known for generic systems. Moreover, for these classes of polynomials, it has been shown in [17] that the degrees of the iterations of the polynomials Fi, i = 1, . . . , m, grow signi\ufb01cantly slower than the exponential growth expected for the iterations of a \u201cgeneric\u201d system of m polynomials in m variables. In turn, this leads (see [18]) to much better estimates of exponential sums, and thus of discrepancy, for vectors generated by (5) than for those originated from arbitrary polynomial systems (see [4, 5, 16]). We also note that the results obtained in [17, 18] regarding the degree growth of the iterations of the polynomials in (5) hold over any \ufb01eld F. In this paper we extend the class of rational dynamical systems with slow degree growth and present an analogue of the construction (5), but with rational functions de\ufb01ned by F1(X1, . . . , Xm) = Xe1 1 G1(X2, . . . , Xm) + H1(X2, . . . , Xm), . . . Fm\u22121(X1, . . . , Xm) = Xem\u22121 m\u22121 Gm\u22121(Xm) + Hm\u22121(Xm), Fm(X1, . . . , Xm) = gmXem m + hm, (6) with e1, . . . , em \u2208{\u22121, 1}, Gi, Hi \u2208F[Xi+1, . . . , Xm], i = 1, . . . , m \u22121, and gm, hm \u2208F, gm \u0338= 0. 3 \fWe note that for m = 1 and e = 1 we obtain the classical linear congruential generator which have been successully used for decades in the theory of Quasi Monte Carlo methods, see [7, 8], and for m = 1 and e = \u22121, the classical inversive generator, see [9, 10, 11, 12]. For the above class of multivariate rational functions, we study the degree growth under iterations and, using an approach similar to that of [17, Lemma 1], we show in Section 3 that under certain additional conditions imposed on the systems of rational functions (6), the degree grows polynomially. Moreover, for applications to pseudorandom number generators, following the standard technique almost identical to that of [17], one almost immediately obtains bounds on the exponential sums with elements of the sequence (4) generated by the system (6) (satisfying the conditions outlined in Section 3), that in turn leads to estimates on the uniformity of distribution of the vectors (4). However, one has also to prove that for any k \u0338= l and nonzero vector a = (a1, . . . , am\u22121) \u2208Fm\u22121, the linear combination Qk,l,a = m\u22121 X i=1 ai(F (k) i \u2212F (l) i ) (7) is a non-constant rational function. We note that in the case of rational functions this does not follow directly from the degree argument as in the case of the polynomial systems (5) in [17], but we give such a result in Section 4. Since the derivation of such bounds of exponential sums for our systems does not bring anything new to the area, we do not do this here but rather concentrate on the study of the degree and linear independence of iterates, which is also of interest for the general area of algebraic dynamics. Furthermore, we consider a related question about the length of trajectories generated by iterations (4) over a \ufb01nite \ufb01eld Fq. We remark that in this case a trajectory falls into a cycle if ut = us for some integers t > s \u22650. In particular, we show that under some rather broad conditions for any \ufb01xed \u03b5 > 0, for all but o(qm) initial vectors u0 \u2208Fm q , the trajectory length t of the iterations (4) is at least q1/3\u2212\u03b5. We note that Silverman [23] has considered a question about periods of general polynomial systems but in somewhat dual situation when the initial value is \ufb01xed and the iterations are considered over a family of \ufb01nite \ufb01elds. 4 \fThe results of [23] apply to very general systems, however the estimates are only logarithmic rather than a power of the \ufb01eld size. Moreover, we give necessary and su\ufb03cient conditions for the systems (6) to generate sequences of maximal period. We note that for the case ei = 1 for all i = 1, . . . , m, the maximal period length of the sequence generated by the system (5) is achieved whenever the conditions of [15, Theorem 6] are satis\ufb01ed. Our result is a generalisation of that of [15]. 2 Structure of the Iterations As in [17], we can describe explicitly the iterations of the rational functions Fi as follows. Let us de\ufb01ne the sets I+ = {1 \u2264i \u2264m : ei = 1} and I\u2212= {1 \u2264i \u2264m : ei = \u22121}. (8) We also de\ufb01ne G(\u2113) i (Xi+1, . . . , Xm) = Gi \u0010 F (\u2113\u22121) i+1 , . . . , F (\u2113\u22121) m \u0011 , H(\u2113) i (Xi+1, . . . , Xm) = Hi \u0010 F (\u2113\u22121) i+1 , . . . , F (\u2113\u22121) m \u0011 . Lemma 1. Let F1, . . . , Fm be rational functions de\ufb01ned by (6). Then for i = 1, . . . , m \u22121 and k = 0, 1, . . ., for the rational functions F (k) i given by (1), 1. for every i \u2208I+, i < m, we have F (k) i = XiGi,k + Hi,k, (9) where Gi,k, Hi,k \u2208F(Xi+1, . . . , Xm) are de\ufb01ned by Gi,k = GiG(2) i . . . G(k) i , Hi,k = HiG(2) i . . . G(k) i + H(2) i G(3) i . . . G(k) i + . . . + H(k\u22121) i G(k) i + H(k) i ; (10) 2. for every i \u2208I\u2212, i < m, we have: F (k) i = XiRi,k + Si,k XiRi,k\u22121 + Si,k\u22121 , (11) 5 \fwhere Ri,k, Si,k are de\ufb01ned by the recurrence relations Ri,k = G(k) i Ri,k\u22122 + H(k) i Ri,k\u22121 Si,k = G(k) i Si,k\u22122 + H(k) i Si,k\u22121 (12) for k \u22651, with the initial rational functions Ri,0 = 1, Si,0 = 0, Ri,1 = Hi, Si,1 = Gi; 3. if m \u2208I+, then F (k) m = gk mXm + (gk\u22121 m + . . . + gm + 1)hm; 4. if m \u2208I\u2212, then F (k) m = (Ak)1,1Xm + (Ak)1,2 (Ak)2,1Xm + (Ak)2,2 , where Ak = \u0012 hm gm 1 0 \u0013k = \u0012 (Ak)1,1 (Ak)1,2 (Ak)2,1 (Ak)2, 2 \u0013 . Proof. The case ei = 1, i = 1, . . . , m, is given by [18, Lemma 1]. We consider now that ei = \u22121 and prove the result by induction on the number of iterations k. For k = 1 it is clear from the de\ufb01nition of the system, so we consider the statement true for the \ufb01rst k \u22121 iterations and we prove it for the k-th iteration. For i = 1, . . . , m \u22121, we have F (k) i = Fi(F (k\u22121) i , F (k\u22121) i+1 , . . . , F (k\u22121) m ) = F (k\u22121) i H(k) i + G(k) i F (k\u22121) i = XiRi,k\u22121+Si,k\u22121 XiRi,k\u22122+Si,k\u22122H(k) i + G(k) i XiRi,k\u22121+Si,k\u22121 XiRi,k\u22122+Si,k\u22122 = Xi(G(k) i Ri,k\u22122 + H(k) i Ri,k\u22121) + G(k) i Si,k\u22122 + H(k) i Si,k\u22121 XiRi,k\u22121 + Si,k\u22121 and thus we conclude this case. When em = \u22121, it is also clear as the k-th iteration of Fm = hmXm + gm Xm is given by Ak as simple calculations show. \u2293 \u2294 6 \fWe want to describe the degree growth of the iterations of the rational functions de\ufb01ned by (6), and in particular to prove that we have the same e\ufb00ect of slow degree growth as for the polynomial systems (5) described in [17, Lemma 1]. To be able to give an explicit formula for the degree growth we need to impose some further conditions on the degrees of the polynomials Gi and Hi, i = 1, . . . , m \u22121. Let F1, . . . , Fm be rational functions de\ufb01ned by (6). From now on we consider the system (6) satisfying the following conditions for Fi for any i = 1, . . . , m: 1. if ei = 1, as in [17, 18], we assume that the polynomial Gi has a unique leading monomial X si,i+1 i+1 . . . X si,m m , that is Gi = giX si,i+1 i+1 . . . Xsi,m m + e Gi, where gi \u2208F\u2217and e Gi \u2208F[Xi+1, . . . , Xm] with degXj e Gi < si,j, degXj Hi < si,j, j = i + 1, . . . , m; (13) 2. if ei = \u22121, we assume that the polynomial Hi has a unique leading monomial X si,i+1 i+1 . . . X si,m m , that is Hi = hiX si,i+1 i+1 . . . Xsi,m m + e Hi, where hi \u2208F\u2217and e Hi \u2208F[Xi+1, . . . , Xm], and degXj e Hi < si,j, degXj Gi < 2si,j, j = i + 1, . . . , m. (14) We note that having these conditions also allows us to consider the rational function system with constant multipliers Gi, i = 1, . . . , m \u22121. We remark that in [14], the case of constant polynomials Gi, i = 1, . . . , m \u22121, in the system (5) was considered, but this case is di\ufb00erent from the case of rational functions as the conditions on the degrees also di\ufb00er, see (13) and (14). Having this, we prove the following formula for the degree growth which coincides with [17, Lemma 1]. 7 \f3 Degree Growth Theorem 2. Let F1, . . . , Fm be rational functions de\ufb01ned by (6) satisfying the conditions (13) and (14) and such that si,i+1 \u0338= 0, i = 1, . . . , m\u22121. Then the degrees of the iterations of F1, . . . , Fm grow as follows deg F (k) i = 1 (m \u2212i)!km\u2212isi,i+1 . . . sm\u22121,m + \u03c8i(k), i = 0, . . . , m \u22121, deg F (k) m = 1, where \u03c8i(T) \u2208Q[T] is a polynomial of degree deg \u03c8i < m \u2212i. Proof. The proof is based on Lemma 1. The case ei = 1, using (9) and (11), follows exactly the same as in [17, Lemma 1]. We prove now the case when ei = \u22121. Using the conditions (13) and (14) and the recurrence relation (12), it is easy to see that deg F (k) i = deg Ri,k + 1 = deg H(k) i Ri,k\u22121 + 1 = deg HiH(2) i . . . H(k) i + 1 = k X j=1 deg H(j) i + 1. (15) As in [17, Lemma 1], we use induction on the number of variables m. For m = 2 we easily see that deg H(j) 1 = deg H1 = s1,2, and thus, by (15), we have that deg F (k) 1 = ks1,2 + 1. We assume now that the theorem is true for m \u22121 variables and we prove it for m. For any i = 1, . . . , m \u22121, by the induction hypothesis, we have deg F (k) i = k X j=1 deg H(j) i + 1 = k X j=1 deg Hi(F (j\u22121) i+1 , . . . , F (j\u22121) m ) + 1 = k X j=1 deg \u0010 (F (j\u22121) i+1 )si,i+1 . . . (F (j\u22121) m )si,m\u0011 + 1 = k X j=1 \u0012 1 (m \u2212i \u22121)!(j \u22121)m\u2212i\u22121si,i+1si+1,i+2 . . . sm\u22121,m+ . . . +(j \u22121)si,msm\u22121,m) + 1. As k X j=1 jm\u22121\u2212i = 1 m \u2212i(Bm\u2212i(k + 1) \u2212Bm\u2212i(0)), 8 \fwhere Bm\u2212i is the Bernoulli polynomial of degree m\u2212i (which has the leading coe\ufb03cient equal to 1), we \ufb01nally obtain the desired result. \u2293 \u2294 4 Linear Independence Theorem 3. Let F1, . . . , Fm be rational functions de\ufb01ned by (6) satisfying the conditions (13) and (14) and such that si,i+1 \u0338= 0, i = 1, . . . , m\u22121. Then, for k \u0338= l and a nonzero vector a \u2208Fm\u22121, Qk,l,a is a non-constant rational function. Proof. The proof reduces to proving that deg(F (k) s \u2212F (l) s ) > 1, where s \u2264 m \u22121 is the smallest index such that as \u0338= 0, as the variable Xs does not appear in the polynomial Qk,l,a \u2212as(F (k) s \u2212F (l) s ) = m\u22121 X i=s+1 ai(F (k) i \u2212F (l) i ). If es = 1, it is clear, as from Theorem 2, for k > l we have deg Gs,k > deg Gs,l. If es = \u22121, by (11), we have F (k) s \u2212F (l) s = XsRs,k + Ss,k XsRs,k\u22121 + Ss,k\u22121 \u2212 XsRs,l + Ss,l XsRs,l\u22121 + Ss,l\u22121 = Uk,l,s Vk,l,s , where Uk,l,s = X2 s(Rs,kRs,l\u22121 \u2212Rs,k\u22121Rs,l) + Xs(Rs,kSs,l\u22121 + Ss,kRs,l\u22121 \u2212Rs,k\u22121Ss,l \u2212Ss,k\u22121Rs,l) + Ss,kSs,l\u22121 \u2212Ss,k\u22121Ss,l. and Vk,l,s = X2 s Rs,k\u22121Rs,k\u22121 + Xs(Rs,k\u22121Ss,l\u22121 + Rs,l\u22121Ss,k\u22121) + Ss,k\u22121Ss,l\u22121. Without loss of generality we may assume that k > l. Using (12), we obtain Rs,kRs,l\u22121 \u2212Rs,k\u22121Rs,l = (G(k) s Rs,k\u22122 + H(k) s Rs,k\u22121)Rs,l\u22121 \u2212Rs,k\u22121(G(l) s Rs,l\u22122 + H(l) s Rs,l\u22121), 9 \fand thus, using Lemma 1 and Theorem 2, we derive deg(Rs,kRs,l\u22121 \u2212Rs,k\u22121Rs,l) = deg(H(k) s \u2212H(l) s )Rs,k\u22121Rs,l\u22121 = deg H(k) s Rs,k\u22121Rs,l\u22121 > deg Rs,k\u22121Rs,l\u22121 > 1 (16) for k > l, which concludes the proof. \u2293 \u2294 Note that, as in [18], we can include m-term linear combinations Qk,l,a = m X i=1 ai(F (k) i \u2212F (l) i ) with a \u2208Fm, but in the case of a1 = . . . = am\u22121 = 0, am \u0338= 0, the nontriviality also depends on the divisibility of k \u2212l by the multiplicative order of gm. 5 Trajectory Lengths In this section we work over a \ufb01nite \ufb01eld Fq. Theorem 4. Let F1, . . . , Fm be rational functions de\ufb01ned by (6) satisfying the conditions (13) and (14) and such that si,i+1 \u0338= 0, i = 1, . . . , m\u22121. Then, for any T \u22651 for all but O(T 3qm\u22121) initial vectors u0 \u2208Fm q , the trajectory length of the iterations (4) exceeds T. Proof. Let U be the set of u = (u1, . . . , um) \u2208Fm q such that Gi(ui+1, . . . , um) = 0 for some i = 1, . . . , m \u22121. Clearly #U = O(qm\u22121). We now build a sequence of sets Uk, k = 0, 1, . . ., recursively. We put U0 = U. Assume that U0, . . . , Uk are de\ufb01ned and let Wk = U0 \u222a. . . \u222aUk. Then we let Uk+1 be the set of the initial values u0 \u2208Fm q \\ Wk such that for the corresponding sequence of vectors (4) we have uk+1 \u2208U. Inspecting (6), we now see that, by our assumption, for any v \u2208Fm q , there 10 \fis a unique preimage u \u2208Fm q \\ U under the map given by (6) (that is, with v = F(u)). In turn, we see that for any v \u2208U there is a unique corresponding initial value u0 \u2208Fm q \\ Wk with uk+1 = v. Thus #Uk = #U. Since there are O(qm\u22121) vectors u \u2208Fm q that contain a zero in at least one component, we see that the set ET of initial values for which, for some integer t \u2264T, the vector ut has a zero component, satis\ufb01es ET = O(qm\u22121 + T#U) = O(Tqm\u22121). (17) We see that if a vector u0 \u2208Fm q \\ ET generates a trajectory of lengths t \u2264T then ut = us for some nonnegative integer s < t. Now, if em\u22121 = 1, then we remove the set FT of initial vectors u0 \u2208Fm q such that Gm\u22121,t(u0) = Gm\u22121,s(u0) for some integers s and t with T \u2265t > s \u22650. By Lemma 1 we see that Gm\u22121,t \u2212Gm\u22121,s is a nontrivial polynomial of degree O(t). Hence, #FT = O   X 0\u2264s<t\u2264T tqm\u22121 ! = O \u0000T 3qm\u22121\u0001 . (18) Furthermore, if em\u22121 = \u22121, then we remove the set FT of initial vectors u0 \u2208Fm q such that Rm\u22121,t(u0)Rm\u22121,s\u22121(u0) = Rm\u22121,t\u22121(u0)Rm\u22121,s(u0) for some integers s and t with T \u2265t > s \u22650. As in the proof of Theorem 3 (in particular, see (16)) we note that Lemma 1 implies that Rm\u22121,tRm\u22121,s\u22121\u2212 Rm\u22121,t\u22121Rm\u22121,s is a nontrivial polynomial of degree O(t). Hence, again we obtain the bound (18). We remark that for T \u2265t > s \u22650, for any solution u0 = (u0,1, . . . , u0,m) \u2208 Fm q \\ FT to the equation F(t)(u0) = F(s)(u0), the component u0,m\u22121 is uniquely de\ufb01ned by u0,m. So there at most qm\u22121 such solutions for every \ufb01xed t and s with T \u2265t > s \u22650 and thus at most T 2qm\u22121 for such t and s. Combining this bound with (17) and (18) we conclude the proof. \u2293 \u2294 11 \fClearly if the map u 7\u2192F(u) is a permutation, as for example, in [13], then all trajectories are purely periodic. So we always have s = 0 in the argument of the proof of Theorem 4. This leads to a better estimate O(T 2qm\u22121) on the number of initial values generating trajectories of length at most T. 6 Maximal Periods In this section we show that the periods of the rational function systems over Fq de\ufb01ned by (6) with ei = \u22121 for all i = 1, . . . , m are given by the orbit lengths of certain linear fractional transformations, also called M\u00a8 obius transformations. In particular, we describe the case when the systems (6) achieve maximal periods in their orbits. We also note that in [15, Theorem 6] there are given necessary and su\ufb03cient conditions for the system (6) to achieve maximal period in the case ei = 1 for all i = 1, . . . , m. We denote e u0,i = (u0,i+1, . . . , u0,m) \u2208Fm\u2212i q , i = 1, . . . , m \u22121. Lemma 5. Let F1, . . . , Fm \u2208Fq[X1, . . . , Xm] be as in (6) with ei = \u22121 for all i = 1, . . . , m. Assume that the sequence generated by the lower m\u2212i rational functions Fi+1, . . . , Fm in Fm\u2212i q is purely periodic with period \u03c4i+1 for some i = 1, . . . , m \u22121. Then we have the following description for the k\u03c4i+1-th iteration of Fi on any initial vector u0 \u2208Fm q : F (k\u03c4i+1) i (u0) = f (k) i (u0,i), k \u22651, (19) where Ri,\u03c4i+1 and Si,\u03c4i+1 are de\ufb01ned by (12) and fi is the M\u00a8 obius transformation in the variable Y , fi(Y ) = Y Ri,\u03c4i+1(e u0,i) + Si,\u03c4i+1(e u0,i) Y Ri,\u03c4i+1\u22121(e u0,i) + Si,\u03c4i+1\u22121(e u0,i). In particular, the orbit length of Fi in u0 is given by the orbit length of fi in u0,i. Proof. We \ufb01rst note that the orbit length of Fi is a multiple of \u03c4i+1. Indeed, let \u03c4i be the orbit length of the system Fi, . . . , Fm in the initial vector u0. Then \u03c4i = lcm (\u03c4i+1, \u03b7i), where \u03b7i is the period ofthe sequence {un,i} de\ufb01ned by the iterations of the polynomial Fi, and thus \u03c4i is a multiple of \u03c4i+1. This 12 \fshows that, in order to describe the period of Fi, . . . , Fm on the initial vector u0, it is enough to consider only k\u03c4i+1-th iterations of Fi. By (11) we have F (\u03c4i+1) i (u0) = u0,iRi,\u03c4i+1(e u0,i) + Si,\u03c4i+1(e u0,i) u0,iRi,\u03c4i+1\u22121(e u0,i) + Si,\u03c4i+1\u22121(e u0,i) = fi(u0,i). (20) Now, F (k\u03c4i+1) i (u0) =F (\u03c4i+1) i \u0010 F ((k\u22121)\u03c4i+1) i (u0), F ((k\u22121)\u03c4i+1) i+1 (u0), . . . , F ((k\u22121)\u03c4i+1) m (u0) \u0011 =F (\u03c4i+1) i \u0010 F ((k\u22121)\u03c4i+1) i (u0), u0,i+1, . . . , u0,m \u0011 = F ((k\u22121)\u03c4i+1) i (u0)Ri,\u03c4i+1(e u0,i) + Si,\u03c4i+1(e u0,i) F ((k\u22121)\u03c4i+1) i (u0)Ri,\u03c4i+1\u22121(e u0,i) + Si,\u03c4i+1\u22121(e u0,i) . (21) To prove (19) we use induction over k. For k = 1 it is clear. We now assume that the statement is true for k \u22121 and we prove it for k. Using (20), (21) and the induction hypothesis, we derive F (k\u03c4i+1) i (u0) = f (k\u22121) i (u0,i)Ri,\u03c4i+1(e u0,i) + Si,\u03c4i+1(e u0,i) f (k\u22121) i (u0,i)Ri,\u03c4i+1\u22121(e u0,i) + Si,\u03c4i+1\u22121(e u0,i) = f (k) i (u0,i), which concludes the proof. \u2293 \u2294 Lemma 6. Let F = {F1, . . . , Fm} be a system of polynomials over Fq de\ufb01ned by (6). Let i = 1, . . . , m such that ei = \u22121 in the system (6). Then we have Ri,kSi,k\u22121 \u2212Ri,k\u22121Si,k = (\u22121)kGiG(2) i . . . G(k) i , where Ri,k, Si,k are de\ufb01ned by (12). Proof. We use induction on k. If k = 1, by (12), we get Ri,1Si,0 \u2212Ri,0Si,1 = \u2212Gi. We assume the statement true for k and we prove it for k + 1. By (12) and the induction hypothesis we have Ri,k+1Si,k \u2212Ri,kSi,k+1 =(G(k+1) i Ri,k\u22121 + H(k+1) i Ri,k)Si,k \u2212Ri,k(G(k+1) i Si,k\u22121 + H(k+1) i Si,k) = \u2212G(k+1) i (Ri,kSi,k\u22121 \u2212Ri,k\u22121Si,k) =(\u22121)k+1GiG(2) i . . . G(k) i G(k+1) i 13 \fand thus we conclude the proof. \u2293 \u2294 As usual, we say that a polynomial f \u2208Fq[X] of degree d \u22651 is primitive if it is the minimal polynomial over Fq of a primitive element of Fqd (that is, an element of multiplicative order qd \u22121), see [6]. Next, we present necessary and su\ufb03cient conditions for the system (6) to achieve maximal period over the prime \ufb01eld Fp. Using [15, Lemma 2] which holds for the functions Fi in the system (6) for which ei = 1, we have the following analogue of [15, Lemma 5] (with an almost identical proof which we do not present here). Lemma 7. Let F = {F1, . . . , Fm} be a system of polynomials over Fp de\ufb01ned by (6). Let the index 1 \u2264i \u2264m such that ei = 1 and assume that the period of the sequence generated by the lower m\u2212i polynomials Fi+1, . . . , Fm in Fm\u2212i p is pm\u2212i and that Gi,pm\u2212i(e u0,i) = 1. Then, for the rational functions Hi,pm\u2212i de\ufb01ned by (10), we have Hi,pm\u2212i(e u0,i) = X v\u2208Fm\u2212i p Ri(v), where Ri \u2261HiG(2) i . . . G(pm\u2212i) i . (22) Now, using Lemma 7 and [15, Theorem 6] for Fi with ei = 1 in the system (6), we have the following result. We recall that the sets I+ and I\u2212are given by (8). Theorem 8. Let F = {F1, . . . , Fm} be a system of polynomials over Fp de\ufb01ned by (6). Then the sequence {un} generated by (4) is purely periodic with period \u03c4 = pm if and only if the following conditions are satis\ufb01ed 1. for every i \u2208I+, i < m, we have Y v\u2208Fm\u2212i p Gi(v) = 1 and X v\u2208Fm\u2212i p Ri(v) \u0338= 0; 2. for every i \u2208I\u2212, i < m, we have: (a) if Ri,pm\u2212i\u22121(u0) = 0, then Ri,pm\u2212i(u0) = Si,pm\u2212i\u22121(u0) and Si,pm\u2212i(u0)Si,pm\u2212i\u22121(u0) \u0338= 0; 14 \f(b) if Ri,pm\u2212i\u22121(u0) \u0338= 0, then X2 \u2212Ri,pm\u2212i(u0) Ri,pm\u2212i\u22121(u0)X \u2212 Q v\u2208Fm\u2212i p G1(v) Ri,pm\u2212i\u22121(u0) is a primitive polynomial over Fp; 3. if m \u2208I+, then gm = 1; 4. if m \u2208I\u2212, then X2 \u2212hmX \u2212gm is a primitive polynomial over Fp. Proof. We prove the result by induction on m. For m = 1, if m \u2208I+, then it\u2019s clear that the period p is achieved if and only if gm = 1 and hm \u2208F\u2217 p. If m \u2208I\u2212, we have F1 = g1X\u22121 1 + h1, which, by [3, Theorem 1], has maximal period p if and only if the polynomial X2 \u2212h1X \u2212g1 is a primitive polynomial over Fp. We assume the statement true for the \ufb01rst m \u22121 variables and we want to prove it for m. Let the sequence {e un,1} = {(un,2, . . . , un,m)} be de\ufb01ned by the last m \u22121 components of the vectors in the sequence {un}. By the induction hypothesis we know that the period e \u03c4 of the sequence {e un,1} is pm\u22121, and taking into account the \ufb01rst remark in the proof of Lemma 5, we see that the period of {e un,1} is of the form kpm\u22121, for some 1 \u2264k \u2264q. Thus, proving the maximality of the period of {un} reduces to proving that k = q. We note that the representation of F (k) 1 given by Lemma 1 does not depend how we choose e2, . . . , em in the functions F2, . . . , Fm, but only the functions G1,k, H1,k or G(k) 1 , H(k) 1 if 1 \u2208I+ or 1 \u2208I\u2212, respectively. Thus, the representation of F (k) 1 given by Lemma 1 is the same regardless if ij \u2208I+ or ij \u2208I\u2212for 2 \u2264j \u2264m. Thus, the case 1 \u2208I+ follows identically as in the proof of [15, Theorem], and we do not repeat it here. We consider now the case 1 \u2208I\u2212. By Lemma 5 we have F (kpm\u22121) 1 (u0) = f (k) 1 (u0,1), k \u22651, where f1(Y ) = Y R1,pm\u22121(e u0,1) + S1,pm\u22121(e u0,1) Y R1,pm\u22121\u22121(e u0,1) + S1,pm\u22121\u22121(e u0,1), 15 \fand thus the maximal period of the sequence generated by the iterations of F1 is given by the case when f1 achieves maximal orbit length in u0,1. We distinguish now two cases. If R1,pm\u22121\u22121(e u0,1) = 0, then we note that S1,pm\u22121\u22121(e u0,1) \u0338= 0, as otherwise f1 = 0. Thus, we have the case of linear generator f1(Y ) = R1,pm\u22121(e u0,1) S1,pm\u22121\u22121(e u0,1)Y + S1,pm\u22121(e u0,1) S1,pm\u22121\u22121(e u0,1). The maximal period is achieved in this case if and only if R1,pm\u22121(e u0,1) S1,pm\u22121\u22121(e u0,1) = 1, S1,pm\u22121(e u0,1) S1,pm\u22121\u22121(e u0,1) \u0338= 0, which is equivalent to R1,pm\u22121(e u0,1) = S1,pm\u22121\u22121(e u0,1), S1,pm\u22121(e u0,1) \u0338= 0. We consider now the case of R1,pm\u22121\u22121(e u0,1) \u0338= 0. We note that in this case f1 achieves maximal period if and only if, under a linear transformation, it has the same property. Taking into account that R1,pm\u22121\u22121(e u0,1) \u0338= 0, we can make the linear transformation Y \u2192R1,pm\u22121\u22121(e u0,1)\u22121Y \u2212R1,pm\u22121\u22121(e u0,1)\u22121S1,pm\u22121\u22121(e u0,1) and obtain the following inversive generator which, by a slightly abuse of notation, we denote also by f1, f1(Y ) = \u2212R1,pm\u22121(e u0,1)S1,pm\u22121\u22121(e u0,1) + R1,pm\u22121\u22121(e u0,1)S1,pm\u22121(e u0,1) R1,m\u22121(e u0,1) Y \u22121 + R1,pm\u22121(e u0,1) R1,pm\u22121\u22121(e u0,1). Applying now Lemma 6 we have R1,pm\u22121(e u0,1)S1,pm\u22121\u22121(e u0,1)\u2212R1,pm\u22121\u22121(e u0,1)S1,pm\u22121(e u0,1) = \u2212G1(e u0,1)G(2) 1 (e u0,1) . . . G(pm\u22121) 1 (e u0,1). Let e F = {F2, . . . , Fm}. Now, as the period induced by e F is pm\u22121, the elements e u0,1, e F(e u0,1), . . . , e F (pm\u22121\u22121)(e u0,1) 16 \fare all the distinct elements of Fm\u2212i p , and thus we obtain G1(e u0,1)G(2) 1 (e u0,1) . . . G(pm\u22121) 1 (e u0,1) = G1(e u0,1)G1( e F(e u0,1)) . . . G1( e F (pm\u22121\u22121)(e u0,1)) = Y v\u2208Fm\u22121 p G1(v). This concludes that f1(Y ) = Q v\u2208Fm\u22121 p G1(v) R1,pm\u22121(e u0,1) Y \u22121 + R1,pm\u22121(e u0,1) R1,pm\u22121\u22121(e u0,1). Applying now [3, Theorem 1], we know that f1 achieves maximal period q if and only if the polynomial X2 \u2212R1,pm\u22121(e u0,1) R1,pm\u22121\u22121(e u0,1)X \u2212 Q v\u2208Fm\u22121 p G1(v) R1,pm\u22121\u22121(e u0,1) is a primitive polynomial over Fp. \u2293 \u2294 Acknowledgment During the preparation of this paper, A. O. was supported in part by the Swiss National Science Foundation Grant PBZHP2\u2013133399 and I. S. by the Australian Research Council Grant DP1092835."
+                },
+                {
+                    "url": "http://arxiv.org/abs/1001.1504v2",
+                    "title": "Pseudorandomness and Dynamics of Fermat Quotients",
+                    "abstract": "We obtain some theoretic and experimental results concerning various\nproperties (the number of fixed points, image distribution, cycle lengths) of\nthe dynamical system naturally associated with Fermat quotients acting on the\nset $\\{0, ..., p-1\\}$. We also consider pseudorandom properties of Fermat\nquotients such as joint distribution and linear complexity.",
+                    "authors": "Alina Ostafe, Igor E. Shparlinski",
+                    "published": "2010-01-10",
+                    "updated": "2010-01-25",
+                    "primary_cat": "math.NT",
+                    "cats": [
+                        "math.NT",
+                        "math.DS",
+                        "11A07; 11L40; 37A45"
+                    ],
+                    "main_content": "Introduction 1.1 Background For a prime p and an integer u with gcd(u, p) = 1 the Fermat quotient qp(u) is de\ufb01ned as the unique integer with qp(u) \u2261up\u22121 \u22121 p (mod p), 0 \u2264qp(u) \u2264p \u22121, and we also de\ufb01ne qp(kp) = 0, k \u2208Z. It is well-known that the p-divisibility of Fermat quotients qp(a) by p has numerous applications, which include the Fermat Last Theorem and squarefreeness testing, see [13, 15, 17, 24]. In particular, the smallest value \u2113p of u \u22651 for which qp(u) \u0338\u22610 (mod p) plays a prominent role in these applications, for which the following estimates are given [5] \u2113p \u2264 \u001a (log p)463/252+o(1) for all p, (log p)5/3+o(1) for almost all p, (where almost all p means for all p but a set of relative density zero), which improve the previous estimates of the form \u2113p = O ((log p)2) of [15, 18, 21, 24]. It is widely believed that \u2113p = 2 for all primes p, except for a very thin set of so called Wieferich primes, which one expects \u2113p = 3 (in particular, it is expected that \u2113p \u22643 for all primes). The behaviour (and even the in\ufb01nitude) of Wieferich primes is still very poorly understood, although several interesting results, relating Wieferich primes to other number theoretic problems are known, see [19, 26, 29]. There are also several results about the distribution of Fermat quotients. For instance, Heath-Brown [20] has proved that the Fermat quotients qp(u) are asymptotically uniformly distributed (after scaling by 1/p and mapping them into qp(u)/p \u2208[0, 1]) for u = M +1, . . . , M +N for any integers M and N \u2265p1/2+\u03b5 for some \ufb01xed \u03b5 and p \u2192\u221e. Note that [20, Theorem 2] gives this only for N \u2265p3/4+\u03b5 but using the full strength of the Burgess bound one can lower this threshold down to h \u2265p1/2+\u03b5, see Lemma 2 below and also [13, Section 4]. It is also shown in [15, Proposition 2.1] that for any integer a the number of solutions to the equation qp(u) = a, 0 \u2264u < p, is at most #{u \u2208{0, . . . , p \u22121} : qp(u) = a} \u2264p1/2+o(1). (1) 2 \fFinally, we also recall several results on congruences involving Fermat quotients, see [3, 9, 31] and references therein. 1.2 Our results Here we consider the dynamical system generated by Fermat quotients. That is, we \ufb01x a su\ufb03ciently large prime p and, for an initial value u0 \u2208{0, . . . , p\u22121} we consider the sequence un = qp(un\u22121), n = 1, 2, . . . . (2) Clearly, there is some t such that ut = uk for some k < t. Then un+t = un+k for any n \u22650. Accordingly, for the smallest value of t with the above condition, we call u0, . . . , ut\u22121 the orbit of the initial value u0. Here we address various questions concerning the sequences generated by (2) such as the number of \ufb01xed points, image size and the \u201ctypical\u201d orbit length. In particular, we compare their characteristics with those expected from random maps, see [14]. All our numerical results support the natural expectation that the map u 7\u2192qp(u) behaves very similar to a random map on the set {0, . . . , p \u22121}. We also investigate their distribution and other characteristics which are relevant to their use as pseudorandom number generators. As we have mentioned, a result of Heath-Brown [20] implies that the fractions qp(u)/p are uniformly distributed for u = M + 1, . . . , M + N, provided that N \u2265p1/2+\u03b5 for some \ufb01xed \u03b5 > 0. However, the method of [20], based on bounds of multiplicative character sums, such as the Polya-Vinogradov and Burgess bounds, see [22, Theorems 12.5 and 12.6], does not seem to apply to studying the distribution of several consecutive elements (as it is essentially equivalent to estimating short sums of multiplicative characters modulo p2 with polynomial arguments). Here we use a di\ufb00erent approach, to study the distribution of points \u0012qp(u) p , . . . , qp(u + s \u22121) p \u0013 , u = M + 1, . . . , M + N, (3) in the s-dimensional cube, which is nontrivial provided that N \u2265p1+\u03b5 for any \ufb01xed real \u03b5 > 0 and integer s \u22651. We also obtain a nontrivial lower bound on the linear complexity of the sequence qp(u) which is also a very important characteristic of any sequence 3 \frelevant to its applications to both cryptography and Quasi-Monte Carlo methods, see [8, 25, 32]. Besides theoretic estimates, we also present results of several numerical tests. Some of these tests are based on a modi\ufb01cation of an algorithm described in [12, 13], which seems to be more computationally e\ufb03cient. We also address some other algorithmic aspects of computation with Fermat quotients. In particular, we give asymptotic estimates of several new algorithms which we design for this purpose. We note that all heuristic predictions concerning various conjectures about Fermat quotinets (for example, the expected number of Wieferich primes up to x as x \u2192\u221e) are based on the assumption of the pseudorandomness of the map u 7\u2192qp(u). Our results provide some theoretic and experimental support to this assumption which seems to be never systematically veri\ufb01ed prior to our work. Finally, motivated by the pseudorandom nature of the map u 7\u2192qp(u), we also discuss some possibilities of using Fermat quotients for designing cryptographically useful hash functions. We remark that Smart and Woodcock [33] have considered iterations of a related function Lp(u) = up \u2212u p (4) in the ring of p-adic integers. However, the setting of [33] (where p is \ufb01xed, for example p = 2) and our settings where p is the main growing parameter are very di\ufb00erent. 1.3 Acknowledgement The authors are very grateful to Sergei Konyagin for his comments which have led to a signi\ufb01cant improvement of the preliminary version of Theorem 10. Thanks also go to Daniel Sutantyo for his help with Magma programs and Tauno Mets\u00a8 ankyl\u00a8 a for his comments and encouragement. During the preparation of this paper, A. O. was supported in part by the Swiss National Science Foundation Grant 121874 and I. S. by the Australian Research Council Grant DP0556431. 4 \f2 Preparations 2.1 General Notation Throughout the paper, p always denotes prime numbers, while k, m and n (in both the upper and lower cases) denote positive integer numbers. For integers a, b and m \u22651 with gcd(b, m) = 1, we write c = a/b rem m for the unique integer c with bc \u2261a (mod m) and 0 \u2264c < m. We also de\ufb01ne ep(z) = exp(2\u03c0iz/p). The implied constants in the symbols \u2018O\u2019, and \u2018\u226a\u2019 may occasionally depend on an integer parameter s and are absolute otherwise. We recall that the notations U = O(V ) and V \u226aU are both equivalent to the assertion that the inequality |U| \u2264cV holds for some constant c > 0. 2.2 Discrepancy and linear complexity Given a sequence \u0393 of N points \u0393 = \b (\u03b3n,1, . . . , \u03b3n,s)N\u22121 n=0 \t (5) in the s-dimensional unit cube [0, 1)s it is natural to measure the level of its statistical uniformity in terms of the discrepancy \u2206(\u0393). More precisely, \u2206(\u0393) = sup B\u2286[0,1)s \f \f \f \f T\u0393(B) N \u2212|B| \f \f \f \f , where T\u0393(B) is the number of points of \u0393 inside the box B = [\u03b11, \u03b21) \u00d7 . . . \u00d7 [\u03b1s, \u03b2s) \u2286[0, 1)s and the supremum is taken over all such boxes, see [11, 23]. Typically the bounds on the discrepancy of a sequence are derived from bounds of exponential sums with elements of this sequence. The relation is made explicit in the celebrated Erd\u00a8 os-Turan-Koksma inequality, see [11, Theorem 1.21], which we present in the following form. 5 \fLemma 1. For any integer H > 1 and any sequence \u0393 of N points (5) the discrepancy \u2206(\u0393) satis\ufb01es the following bound: \u2206(\u0393) = O \uf8eb \uf8ed1 H + 1 N X 0<|h|\u2264H s Y j=1 1 |hj| + 1 \f \f \f \f \f N\u22121 X n=0 exp   2\u03c0i s X j=1 hj\u03b3n,j !\f \f \f \f \f \uf8f6 \uf8f8, where the sum is taken over all integer vectors h = (h1, . . . , hs) \u2208Zs with |h| = maxj=1,...,s |hj| < H. Finally, we recall that the linear complexity L of an N-element sequence s0, . . . , sN\u22121 in a ring R is de\ufb01ned as the smallest L such that su+L = cL\u22121su+L\u22121 + . . . + c0su, 0 \u2264u \u2264N \u2212L \u22121, for some c0, . . . , cL\u22121 \u2208R, see [8, 25, 32]. 2.3 Exponential sums First, we recall the bound of Heath-Brown [20] on exponential sums with qp(u). Although here we use it only with \u03bd = 2 (exactly as it is given in [20]) we formulate it in full generality. As we have mentioned, the method of Heath-Brown [20] combined with the Polya-Vinogradov bound (when \u03bd = 1) and the Burgess bound (when \u03bd \u22652), see [22, Theorems 12.5 and 12.6], implies the following generalisation of [20, Theorem 2]: Lemma 2. For any \ufb01xed integer \u03bd \u22651, we have max gcd(a,p)=1 \f \f \f \f \f M+N X u=M+1 ep (aqp(u)) \f \f \f \f \f \u226aN1\u22121/\u03bdp(\u03bd+1)/2\u03bd2+o(1), as p \u2192\u221e, uniformly over M and N \u22651. We now recall the following well-known bound, see [22, Bound (8.6)]. Lemma 3. For any integers K and r, we have K\u22121 X k=0 ep(kr) \u226amin \u001a K, p \u2225r\u2225 \u001b , where \u2225r\u2225= min s\u2208Z |r \u2212sp| is the distance between r and the closest multiple of p. 6 \f2.4 Basic properties of Fermat quotients Most of our results are based on the following two well-known properties of Fermat quotients. For any integers k, u and v with gcd(uv, p) = 1 we have qp(uv) \u2261qp(u) + qp(v) (mod p) (6) and qp(u + kp) \u2261qp(u) \u2212ku\u22121 (mod p), (7) see, for example, [13, Equations (2) and (3)]. 3 Dynamical Properties 3.1 Computation of qp(u) As we have mentioned, computing each individual value of qp(u) can be done in O(log p) arithmetic operations on O(log p)-bit integers via repeated squaring computation of up\u22121 modulo p2, we refer to [16] for a background on modular arithmetic and complexity of various algorithms. In particular, one can easily reformulate our complexity estimates in terms of bit operations. Thus computing all values of qp(u), 0 \u2264u < p, requires O(p log p) arithmetic operations on O(log p)-bit integers. Such computation is necessary, for example, to \ufb01nd all \ufb01xed points of the map u 7\u2192qp(u) or for \ufb01nding the image size. Here we show that there is a slightly more e\ufb03cient algorithm which is based on (6) and (7). We assume that we are given a primitive root g modulo p. This can be done at the pre-computation stage and we keep it outside of the algorithm (in any case, it can be found in p1/4+o(1) arithmetic operations on O(log p)-bit integers, see [27], which is lower than the remaining parts of the algorithm). Algorithm 4 (Generating qp(u), 0 \u2264u \u2264p \u22121). Input: A prime p and a primitive root g modulo p with 1 < g < p. Output: A permuted sequence of the values qp(u), 0 \u2264u \u2264p \u22121. 7 \f1. Set qp(0) = 0 and qp(1) = 0. 2. Compute qp(g) using the repeated squaring modulo p2. 3. Set b1 = g and c1 = g\u22121 rem p. 4. For i = 2, . . . , p \u22122 compute (a) bi = gbi\u22121 rem p and ci = ci\u22121g\u22121 rem p; (b) ki = (gbi\u22121 \u2212bi)/p; (c) qp(bi) = qp(g) + qp(bi\u22121) + kici rem p. Theorem 5. Algorithm 4 computes every value qp(u), 0 \u2264u < p \u22121, in O (p) arithmetic operations on O(log p)-bit integers. Proof. The complexity estimate is immediate. The correctness of the algorithm follows from the congruences qp(bi) \u2261qp(gbi\u22121 \u2212kip) \u2261 qp(gbi\u22121) + ki(gbi\u22121)\u22121 \u2261 qp(g) + qp(bi\u22121) + kici (mod p), which in turn follow from (6) and (7). \u2293 \u2294 Note that the algorithm of [12, 13] is very similar, except that it uses g = 2 instead of a primitive root. This makes each step faster, but if 2 is not a primitive root modulo p requires going trough all conjugacy classes of the group generated by 2 modulo p and thus requires more \u201cadministration\u201d of data and also more memory. Unfortunately Algorithm 4 does not help to compute qp(u) for a given value of u unless all values qp(v), 0 \u2264v \u2264p \u22121, are precomputed and stored in a table, after which qp(u) can simple be read from there. We now describe a trade-o\ufb00algorithm which requires less memory but the computation of qp(u) is more expensive than the simple table look-up. It depends on a parameter z \u22652, which can be adjusted to particular algorithmic needs. For a real V < p we use Qp(V ) to denote the table of the values of qp(v) with v \u2208[0, V ]. We see from Theorem 5 that Qp(V ) can be computed in O (min{p, V log p}) arithmetic operations on O(log p)-bit integers. Furthermore, for an integer m, we use Im(V ) to denote the table of the values v\u22121 rem m with v \u2208[1, V ] and gcd(v, m) = 1. Since by the 8 \fEuler theorem v\u22121 \u2261v\u03d5(m)\u22121 (mod m), where \u03d5(m) is the Euler function, we see that Im(V ) can be computed in O (V log m) arithmetic operations on O(log m)-bit integers (there are even more e\ufb03cient modular inversion algorithms with a better bound on the number of bit operations, see [16]; however using them does not change the overall complexity of our algorithm). Algorithm 6 (Computing qp(u) for a given u \u2208[0, p \u22121]). Input: A prime p, a real z \u22652, the tables Qp(p/z), Ip(p/z), Ip2(z) and an integer u \u2208{0, . . . , p \u22121}. Output: The value of qp(u). 1. If u = 0 set qp(u) = 0. 2. Find integers v and w with u \u2261v/w (mod p) and such that 1 \u2264v \u2264 2p/z and |w| \u2264z. 3. Recall r = w\u22121 rem p2 if w > 0 or r = \u2212((\u2212w)\u22121 rem p2) if w < 0 from the table Ip2(z). 4. Compute s with s \u2261v/w (mod p2) and such that 0 \u2264s < p2. 5. Compute k = (s \u2212u)/p. 6. Recall r = v\u22121 rem p from the table Ip(p/z). 7. Recall qp(v) and qp(w) from the table Qp(p/z). 8. Compute qp(u) = (qp(v) \u2212qp(w) + krw) rem p. Theorem 7. For any integer u with 0 \u2264u < p \u22121, Algorithm 6 computes qp(u) in O (log z) arithmetic operations on O(log p)-bit integers. Proof. The correctness of the algorithm follows from the congruences qp(u) \u2261 qp(s \u2212kp) \u2261qp(s) + ks\u22121 \u2261 qp(v) \u2212qp(w) + kv\u22121w \u2261qp(v) \u2212qp(w) + krw (mod p) which in turn follow from (6) and (7). 9 \fIt remains to estimate the complexity of \ufb01nding the v and w with u \u2261v/w (mod p). We can also assume that z < p since otherwise the result is trivial. We start computing continued fraction convergents ai/bi, gcd(ai, bi) = 1, i = 1, 2, . . ., to u/p, see, for example, [30] for basic properties of continued fractions. We de\ufb01ne j by the condition bj \u2264z < bj+1. By the well-known property of continued fractions, we have \f \f \f \f aj bj \u2212u p \f \f \f \f \u2264 1 bjbj+1 \u22641 bjz. We now de\ufb01ne w = |ajp \u2212bju| and note that (since z < 0) 0 < w = bjp \f \f \f \f aj bj \u2212u p \f \f \f \f \u2264p z. Furthermore uv \u2261w (mod p) for either v = aj or v = \u2212aj. Finally, since the denominators of the convergents grow at least exponentially, we see that j = O(log bj) = O(log z) and thus \ufb01nd aj and bj in O(log z) steps, each of them requires to compute with O(log p)-bit integers. \u2293 \u2294 We see from Theorem 7 taken with z = exp \u0000\u221alog p \u0001 , that evaluating (in time p exp \u0000\u2212(1 + o(1))\u221alog p \u0001 ) and storing p exp \u0000\u2212(1 + o(1))\u221alog p \u0001 values of Fermat quotients, we can compute any other value in time (log p)1/2+o(1). 3.2 Fixed Points Let F(p) denote the number of \ufb01xed points of the map qp(u) that is, F(p) = #{u \u2208{0, . . . , p \u22121} : qp(u) = u}. We derive a nontrivial estimate on F(p) from Lemmas 1 and 2 Theorem 8. We have F(p) \u226ap11/12+o(1) as p \u2192\u221e. 10 \fProof. Let us choose some positive integer parameter N \u2208[1, p\u22121] and for an integer M we denote by T(p; M, N) the number of integers u \u2208[M+1, M+N] with qp(u) \u2208[M + 1, M + N]. Considering the discrepancy of the fractions qp(u)/p, u = M + 1, . . . , M + N and combining Lemma 1 (taken with s = 1) with Lemma 2 (taken with \u03bd = 2) , we immediately conclude T(p; M, N) = N2 p + O \u0000N1/2p3/8+o(1)\u0001 . Clearly every u = M + 1, . . . , M + N which is a \ufb01xed point contributes to T(p; M, N). Covering the interval [0, p \u22121] with at most (p/N + 1) intervals of length h we obtain F(p) \u2264 \u0010 p N + 1 \u0011 \u0012N2 p + O \u0000N1/2p3/8+o(1)\u0001\u0013 . Choosing N = \u0006 p11/12\u0007 , we conclude the proof. \u2293 \u2294 There is little doubt that the bound of Theorem 8 is very imprecise. It is easy to see that in the full range 0 \u2264u \u2264p2 \u22121 the relation (7) implies #{u \u2208{0, . . . , p2 \u22121} : qp(u) \u2261u (mod p)} = 2p \u22121. Indeed, it is enough to write u = v + kp with v, k \u2208{0, . . . , p \u22121} and notice that \u2022 either v = 0 and then k can take any values \u2022 or v > 0 and then the relation (7) identify k uniquely. Thus one can expect that F(p) = O(1). In fact it seems reasonable to expect that the map u 7\u2192qp(u) behaves similar to a random map. We recall that for a random map on m elements, the probability of having k \ufb01xed points is 1 mm \u0012m k \u0013 \u00d7 (m \u2212k \u22121)m\u2212k \u21921 ek! as m \u2192\u221e. Below we present numerical results giving the numbers N(k) of primes p \u2208[50000, 200000] for which the map u 7\u2192qp(u) has exactly F(p) = k \ufb01xed points (note that we discard the \u201carti\ufb01cial\u201d \ufb01xed point u = 0). We 11 \falso give the proportions of such primes \u03c1(k) = N(k)/N where N = 12851 is the total number of primes p \u2208[50000, 200000] and compare them with \u03c10(k) = (ek!)\u22121 for k = 0, . . . , 6. We note that in the above range N(k) = 0 for k \u22657. k 0 1 2 3 4 5 6 \u03c10(k) 0.368 0.368 0.184 0.0613 0.0153 0.00306 0.000511 N(k) 4770 4697 2327 844 174 36 3 \u03c1(k) 0.371 0.365 0.181 0.0656 0.0135 0.00280 0.000233 Statistics of \ufb01xed points These numerical results appear to indicate a reasonable agreement between the prediction and actual results. 3.3 Concentration of values For integers k and h \u22651 we denote by U(p; k, h) the number of u \u2208{0, . . . , p\u2212 1} for which qp(u) \u2261z (mod p) for some z \u2208[k + 1, k + h]. As in the proof of Theorem 8, a combination of Lemma 2 (which we take with N = p and \u03bd = 2) with Lemma 1 gives the following asymptotic formula U(p; k, h) = h + O(p7/8+o(1)) (8) as p \u2192\u221e. On the other hand, using (1), we trivially obtain U(p; k, h) \u2264hp1/2+o(1) that improves (8) for h \u2264p3/8. We now obtain a better upper bound, which improves (8) for h \u2264p3/4. Theorem 9. For any integers k and h \u22651, we have U(p; k, h) \u2264h1/2p1/2+o(1) as p \u2192\u221e. Proof. Let U be the set of u \u2208{0, . . . , p\u22121}, which are counted by U(p; k, h). Using (6) we see that any w of the form w = uv with uv \u2208U satis\ufb01es 0 \u2264w \u2264p2 \u22121 and qp(w) \u2261z (mod p) (9) 12 \ffor some z \u2208[2k + 2, 2k + 2h]. For a \ufb01xed integer z, there are O(p) values of w \u2208{0, . . . , p2 \u22121} satisfying (9), which follows immediately from (7) (see also the proof of [15, Proposition 2.1]). So there are at most O(hp) values of w satisfying (9) with some z \u2208[2k + 2, 2k + 2h]. Using the classical estimate \u03c4(w) = wo(1), w \u2192\u221e, on the divisor function \u03c4(w) (see [22, Bound (1.81)] with k = 2), we deduce that each w = uv can be obtained from no more than po(1) distinct pairs (u, v) \u2208U2. Therefore (#U)2 \u2264hp1+o(1), which concludes the proof. \u2293 \u2294 3.4 Image size Let M(p) be the image size of the qp(u) for 0 \u2264u \u2264p \u22121, that is M(p) = #{qp(u) : 0 \u2264u \u2264p \u22121}. The bound (1) immediately implies M(p) \u2265p1/2+o(1). In fact more precise bounds \u221ap \u22121 \u2264M(p) \u2264p \u2212 p (p \u22121)/2 can be obtained from (6) and (7), see [13, Section 3]. We now obtain a stronger lower bound on M(p). Theorem 10. We have M(p) \u2265(1 + o(1)) p (log p)2, as p \u2192\u221e. Proof. Let Q(p, a) be the number of primes \u2113\u2208{1, . . . , p \u22121} with qp(\u2113) = a (note that we have discarded u = 0). Clearly p\u22121 X a=0 Q(p, a) = \u03c0(p \u22121) (10) where, as usual, \u03c0(x) denotes the number of primes \u2113\u2264x, and also p\u22121 X a=0 Q(p, a)2 = #R(p), (11) 13 \fwhere R(p) = {(\u2113, r) : 1 \u2264\u2113, r \u2264p \u22121, \u2113, r primes qp(\u2113) = qp(r)}. We see from (6) that if (\u2113, r) \u2208R(p) and w \u2261\u2113/r (mod p2) (12) then qp(w) \u2261qp(\u2113) \u2212qp(r) \u22610 (mod p). Since all w with qp(w) \u22610 (mod p) and gcd(w, p) = 1 have wp\u22121 \u22611 (mod p2), they are elements of the group Gp of the pth power residues modulo p. Thus we see from (12) that #R(p) \u2264N(p), where N(p) is the number of solutions (\u2113, r, w) to w\u2113\u2261r (mod p2), where \u2113, r \u2264p \u22121, \u2113, r primes, w \u2208Gp. (13) We note that for w \u22611 (mod p2) there are exactly \u03c0(p \u22121) pairs (\u2113, r) with \u2113= r that satisfy (13). For any other w \u2208Gp if (13) is satis\ufb01ed for (\u21131, r1) and (\u21132, r2) then \u21131r2 \u2261\u21132r1 (mod p2) which in turn implies the equation \u21131r2 = \u21132r1 (14) (since 1 \u2264\u21131, \u21132r1, r2 \u2264p \u22121). Because \u21131, \u21132r1, r2 are primes, we see from (14) that either (\u21131, \u21132) = (r1, r2), which is impossible for w \u0338\u22611 (mod p2), (\u21131, r1) = (\u21132, r2), which means that when w \u2208Gp \\ {1} is \ufb01xed, then (13) is satis\ufb01ed for at most one pair of primes (\u2113, r). Therefore #R(p) \u2264N(p) \u2264\u03c0(p \u22121) + #Gp \u22121 = p + O(p/ log p). (15) Now, since by the Cauchy inequality we have  p\u22121 X a=0 Q(p, a) !2 \u2264M(p) p\u22121 X a=0 Q(p, a)2, 14 \frecalling (10) and (11) and using (15), we obtain M(p) \u2265(1 + o(1))\u03c0(p \u22121)2p\u22121. which concludes the proof. \u2293 \u2294 Clearly the bound of Theorem 10 is not tight. The image size Mm of a random map on an m element set is expected to be Mm = \u0012 1 \u22121 e \u0013 m = 0.63212 . . .m see [14, Theorem 2], and thus it is reasonable to expect that M(p)/p \u22481\u22121/e. We now give the average value of M(p)/p taken over primes p in the intervals Ji = [50000i, 50000(i + 1)], i = 1, 2, 3. (16) and the whole interval J = [50000, 200000]. (17) Range J1 J2 J3 J # of primes 4459 4256 4136 12851 M(p)/p 0.63212 0.63208 0.63212 0.63211 Statistics of image sizes 3.5 Distribution of orbit lengths For any map f de\ufb01ned on an m element set, and any initial value u0 from this set, we consider the iterations ui = f(ui\u22121), i = 1, 2, . . .. Then for some \u03c1 > \u00b5 \u22650 we have u\u03c1 = u\u00b5. The smallest value of \u03c1 is called the orbit length and the corresponding (and thus uniquely de\ufb01ned) value of \u00b5 is called the tail length. By [14, Theorem 3] the expected values \u03c1m and \u00b5m of the orbit and tail length, taken over all random maps and initial values u0, satisfy \u03c1m \u221am = p \u03c0/2 + o(1) and \u00b5m \u221am = p \u03c0/8 + o(1), as m \u2192\u221e. Here we present the results of computation of the average values of the orbit and the tail lengths, scaled by \u221ap, for the sequence (2) taken over primes p in the intervals J1, J2, J3 and J , given by (16) and (17), respectively, and a randomly chosen initial value u0 \u2208[1, p \u22121]. 15 \fRange J1 J2 J3 J # of primes 4459 4256 4136 12851 \u03c1/\u221ap 1.2423 1.2445 1.2444 1.2437 \u00b5/\u221ap 0.62179 0.62200 0.61806 0.62066 Statistics of orbit and the tail lengths, random u0 Since the values qp(2) are of special interest, we also present similar data where the inutial value is alway chosen as u0 = 2. Range J1 J2 J3 J # of primes 4459 4256 4136 12851 \u03c1/\u221ap 1.2381 1.2507 1.2401 1.2429 \u00b5/\u221ap 0.61778 0.63004 .62060 0.62275 Statistics of orbit and the tail lengths, u0 = 2 The results show quite satisfactory matching with the expected values of p \u03c0/2 = 1.2533 . . . and p \u03c0/8 = 0.62665 . . .. Furthermore, we also give similar average values for C(p)/p, where C(p) is the total number of cyclic points in all possible trajectories of the map u 7\u2192qp(u) on the set {0, . . . , p\u22121}, taken over primes from the same intervals J1, J2, J3 and J . Range J1 J2 J3 J # of primes 4459 4256 4136 12851 C(p)/\u221ap 1.2413 1.2527 1.23706 1.2437 Statistics of cyclic points By [14, Theorem 2] the number Cm of cyclic nodes of a random map on an m element set is expected to be Cm = p \u03c0/2m = 1.2533 . . . , which again is very close to the observed average values. 16 \f4 Pseudorandomness 4.1 Joint distribution For integers M, N \u22651, s \u22651 and an integer vector a = (a0, . . . , as\u22121) we consider the exponential sums Ss,p(M, N; a) = M+N X u=M+1 ep  s\u22121 X j=0 ajqp(u + j) ! . Thus the above sums are generalisations of those of Lemma 2 that correspond to the case s = 1. However the method of Heath-Brown [20] does not seem to apply to the sums Ss,p(M, N; a) as it requires good estimates of mulitiplicative character sums with polynomials, which are not currently known (see however [6] for some potential approaches in the case s = 2). We are now ready to prove an estimate on Ss,p(M, N; a) which together with Lemma 1 implies an upper bound on the discrepancy of points (3). Theorem 11. For any integer s \u22651, we have max gcd(a0,...,as\u22121,p)=1 |Ss,p(M, N; a)| \u226asp log p uniformly over M and p2 > N \u22651. Proof. Select any a = (a0, . . . , as\u22121) \u2208Zs with gcd(a0, . . . , as\u22121, p) = 1 and take K = \u230aN/p\u230b. We get Ss,p(M, N; a) = M+Kp X u=M+1 ep  s\u22121 X j=0 ajqp(u + j) ! + O(p) = Kp X u=1 ep  s\u22121 X j=0 ajqp(u + M + j) ! + O(p) = p X v=1 K\u22121 X k=0 ep  s\u22121 X j=0 ajqp(v + M + j + kp) ! + O(p). Let V be the set of v = 1, . . . , p with v \u0338\u2261\u2212M \u2212j (mod p) for any j = 0, . . . , s \u22121. Therefore, using (7), we obtain: Ss,p(M, N; a) = W + O(p + sK), (18) 17 \fwhere W = X v\u2208V K\u22121 X k=0 ep  s\u22121 X j=0 (ajqp(v + M + j) \u2212ajk(v + M + j)\u22121) ! = X v\u2208V ep  s\u22121 X j=0 ajqp(v + M + j) ! K\u22121 X k=0 ep   \u2212k s\u22121 X j=0 aj(v + M + j)\u22121) ! . Taking now the absolute value, we obtain |W| \u2264 X v\u2208V \f \f \f \f \f K\u22121 X k=0 ep   k s\u22121 X j=0 aj(v + M + j)\u22121) !\f \f \f \f \f . Recalling Lemma 3, we deduce |W| \u2264 X v\u2208V min \u001a K, p \u2225Fa,s(v)\u2225 \u001b , where Fa,s(V ) = s\u22121 X j=0 aj V + M + j . Examining the poles of Fa,s(v), we see that if gcd(a0, . . . , as\u22121, p) = 1 then it is a nonconstant rational function of degree O(s) modulo p. Thus every residue modulo p occurs O(s) times among the values Fa,s(v), v \u2208V. Hence |W| \u226as p\u22121 X u=0 min \u001a K, p \u2225u\u2225 \u001b \u226asp log p which concludes the proof. \u2293 \u2294 Using Lemma (1), we immediately obtain: Corollary 12. For any \ufb01xed s, the discrepancy \u2206p,s(M, N) of points (3) satis\ufb01es \u2206p,s(M, N) \u226aN\u22121p(log p)s+1, uniformly over M and p2 > N \u22651. 18 \f4.2 Linear complexity Here we estimate the linear complexity for a su\ufb03ciently long sequence of consecutive values of qp(u). Theorem 13. For p2 > N \u22651 the linear complexity Lp(N) of the sequence qp(u), u = 0, . . . , N \u22121, satis\ufb01es Lp(N) \u22651 2 min{p \u22121, N \u2212p \u22121}. Proof. Assume that L X j=0 cjqp(u + j) \u22610 (mod p), 0 \u2264u \u2264N \u2212L \u22121, (19) for some integers c0, . . . , cL\u22121 and cL = \u22121. Let R = min{p \u2212L, N \u2212L \u2212p}. Then we see from (19) that for 1 \u2264u \u2264R \u22121 we have L X j=0 cjqp(u + p + j) \u22610 (mod p). (20) Recalling (7) and using (19) again, we now see that L X j=0 cjqp(u + p + j) \u2261 L X j=0 cj \u0000qp(u + j) \u2212(u + j)\u22121\u0001 \u2261\u2212 L X j=0 cj(u + j)\u22121 (mod p). (21) Comparing (20) and (21) we see that L X j=0 cj(u + j)\u22121 \u22610 (mod p), 1 \u2264u \u2264R \u22121. We can assume that L < p since otherwise there is nothing to prove. Clearing the denominators, we obtain a nontrivial polynomial congruence L X j=0 cj L Y h=0 h\u0338=j (u + h) \u22610 (mod p), 19 \fof degree L, which has R\u22121 solutions (to see that it is nontrivial it is enough to substitute u = 0 in the polynomial on the left hand side). Therefore L \u2265R \u22121 and the result follows. \u2293 \u2294 The argument used in the proof of Theorem 13 can also be used to estimate the linear complexity of arbitrary segments of the sequence qp(u), although the resulting bound is slightly weaker. Theorem 14. For M and p2 > N \u22651 the linear complexity Lp(M; N) of the sequence qp(u), u = M + 1, . . . , M + N, satis\ufb01es Lp(M; N) \u2265min \u001ap \u22121 2 , N \u2212p \u22121 3 \u001b . Proof. Assume that L X j=0 cjqp(u + M + j) \u22610 (mod p), 1 \u2264u \u2264N \u2212L, (22) for some integers c0, . . . , cL\u22121 and cL = \u22121. Let R = min{p, N \u2212L \u2212p}. Then we see from (22) that for 1 \u2264u \u2264R we have L X j=0 cjqp(u + M + p + j) \u22610 (mod p). (23) Recalling (7) and using (22) again, we now see that for any integer u with u \u0338\u2261\u2212M \u2212j (mod p), j = 0, . . . , L, we have L X j=0 cjqp(u + M + p + j) \u2261 L X j=0 cj \u0000qp(u + M + j) \u2212(u + M + j)\u22121\u0001 \u2261\u2212 L X j=0 cj(u + M + j)\u22121 (mod p). (24) Comparing (23) and (24) we see that L X j=0 cj(u + M + j)\u22121 \u22610 (mod p), 20 \ffor at least R \u2212L \u22121 values of u with 1 \u2264u \u2264R and u \u0338\u2261\u2212M \u2212j (mod p), j = 0, . . . , L. As before we can assume that L < p since otherwise there is nothing to prove. Clearing the denominators, we obtain a nontrivial polynomial congruence L X j=0 cj L Y h=0 h\u0338=j (u + M + h) \u22610 (mod p) of degree L, which has at least R\u2212L\u22121 solutions (to see that it is nontrivial it is enough to substitute u = \u2212M in the polynomial on the left hand side). Therefore L \u2265R \u2212L \u22121 and the result follows. \u2293 \u2294 5 Hash Functions from Fermat Quotients 5.1 General Construction In this section we propose a new construction of hash functions based on iterations of Fermat quotients. A similar idea, however based on a very di\ufb00erent family of functions, has been previously introduced by D. X. Charles, E. Z. Goren and K. E. Lauter [7]. Let n and r be two positive integers. Choose 2r random (n+1)-bit primes p0, . . . , p2r\u22121. We also consider a random initial n bit integer u0. The has function is built from a sequence of iterations of Fermat quotients moduli p0, . . . , p2r\u22121. As in [7], the input of the hash function is used to decide what modulo what prime the next Fermat quotient is computed. More precisely, given an input bit string \u03a3, we perform the following steps: \u2022 Pad \u03a3 with at most r \u22121 zeros on the left to make sure that its length L is a multiple of r. \u2022 Split \u03a3 into blocks \u03c3j, j = 1, . . . , J, where J = L/r, of length r and interpret each block as an integer \u2113\u2208[0, 2r \u22121]. \u2022 Starting at the point u0, apply the Fermat quotient maps qp\u2113iteratively by using n least signi\ufb01cant bits of uj\u22121 to form an n-bit integer wj\u22121 and then computing uj = qp\u2113(wj\u22121). 21 \f\u2022 Output the last element in the above sequence, that is, uJ = qpJ(wJ\u22121) and outputing its n least signi\ufb01cant bits as the value of the hash function. 5.2 Collision Resistance We remark that the initial element u0 is \ufb01xed and in particular, does not depend on the input of the hash function. Furthermore, the collision resistance is based on the di\ufb03culty of making the decision which Fermat quotient to apply at each step when one attempts to back trace from a given output to the initial element u0 and thus produce two distinct strings \u03a31 and \u03a32 of the same length L, with the same output. Note that for strings of di\ufb00erent lengths, say of L and L+1, a collision can easily be created. It is enough to take \u03a32 = (0, \u03a31) (that is, \u03a32 is obtained from \u03a31 by augmenting it by 0). If L \u0338\u22610 (mod r) then they lead to the same output. Certainly any practical implementation has to take care of things like this. We also note that the results of Section 4 suggest that the above hash functions exhibit rather chaotic behaviour, which close to the behaviour of a random function. It is probably too early to make any suggestions about the applicability of Fermat quotients for hashing but this direction de\ufb01nitely deserves further studying, experimentally and theoretically. 6 Comments Unfortunately we are not able to give any estimates on the discrepancy or linear complexity of the orbits (2), which is a very interesting but possibly hard, question. Obtaining analogues of Theorems 11, 13 and 14, which are nontrivial for N < p is another interesting question. The method of proof of Theorems 13 and 14 does not apply to the nonlinear complexity. We recall the nonlinear complexity of degree d of an N-element sequence s0, . . . , sN\u22121 of elements in a ring R is the smallest L o such that su+L = \u03c8(su+L\u22121, . . . , su), 0 \u2264u \u2264N \u2212L \u22121, 22 \fwhere \u03c8 \u2208R[Y1, . . . , YL] is a polynomial of total degree at most d. Estimating the nonlinear complexity of Fermat quotients is of ultimate interest. Finally, we remark that one can also study the sums Tp(M, N; \u03c7) = M+N X u=M+1 \u03c7 (qp(u)) with a nonprincipal multiplicative character \u03c7 modulo p. Arguing as in the proof of Theorem 11 we get |Tp(M, N; \u03c7)| \u226a M+p\u22121 X v=M+1 \f \f \f \f \f K\u22121 X k=0 \u03c7 \u0000qp(v + M) \u2212k(v + M)\u22121) \u0001 \f \f \f \f \f + p, where K = \u230aN/p\u230b. One can now apply the Burgess bound, see [22, Theorems 12.6], and get a nontrivial estimate on Tp(M, N; \u03c7), starting with N \u2265p5/4+\u03b5 for any \ufb01xed \u03b5 > 0, see [28]. However it is natural to expect that one can take advantage of additional averaging over v and get a nontrivial bound for smaller values of N. Furthermore, using (6) it is possible to estimate bilinear character sums Wp(A, B, U, V ; \u03c7) = X 0\u2264u\u2264U X 0\u2264v\u2264V \u03b1u\u03b2v\u03c7 (qp(uv)) with arbitrary complex weights A = (\u03b1u) and B = (\u03b2v), and then using the Vaughan identity, see [22, Section 13.4], estimate the character sums with Fermat quotients at primes arguments, see [28] for details. Furthermore, we remark that studying the map x 7\u2192(xp\u22121 \u22121)/p in the \ufb01eld of p-adic numbers, is also of great interest, see [33] where a similar question is considered for the maps given by (4). The other way around, it is also quite natural to study the map (4) modulo p. Finally, analogues of Fermat quotients modulo a composite number is certainly an exciting object of study with its own twists, see [1, 2, 4, 10]."
+                },
+                {
+                    "url": "http://arxiv.org/abs/0909.3972v3",
+                    "title": "On the Length of Critical Orbits of Stable Quadratic Polynomials",
+                    "abstract": "We use the Weil bound of multiplicative character sums together with some\nrecent results of N. Boston and R. Jones, to show that the critical orbit of\nquadratic polynomials over a finite field of $q$ elements is of length\n$O(q^{3/4})$, improving upon the trivial bound $q$.",
+                    "authors": "Alina Ostafe, Igor E. Shparlinski",
+                    "published": "2009-09-22",
+                    "updated": "2009-09-28",
+                    "primary_cat": "math.NT",
+                    "cats": [
+                        "math.NT",
+                        "11L40; 11T06; 37F10"
+                    ],
+                    "main_content": "Introduction Let Fq be a \ufb01nite \ufb01eld of q elements. For a polynomial f \u2208Fq[X] we de\ufb01ne the sequence of iterations: f (0)(X) = X, f (n)(X) = f \u0000f (n\u22121)(X) \u0001 , n = 1, 2, . . . . Following [1, 2, 5, 6], we say that f is stable if all polynomials f (n) are irreducible over Fq. We now assume that q is odd. 1 \fAs in [6], for a quadratic polynomial f(X) = aX2 + bX + c \u2208Fq[X], a \u0338= 0, we de\ufb01ne \u03b3 = \u2212b/2a as the unique critical point of f (that is, the zero of the derivative f \u2032) and consider the set Orb(f) = {f (n)(\u03b3) : n = 2, 3, . . .} which is called the critical orbit of f. Clearly there is some t such that f (t)(\u03b3) = f (s)(\u03b3) for some positive integer s < t. Then f (n+t)(\u03b3) = f (n+s)(\u03b3) for any n \u2a7e0. Accordingly, for the smallest value of tf with the above condition, we have Orb(f) = {f (n)(\u03b3) : n = 2, . . . , tf} and #Orb(f) = tf \u22121 or #Orb(f) = tf \u22122 (depending whether s = 1 or s \u2a7e2 in the above). It is shown in [4, 5, 6] that critical orbits play a very important role in the dynamics of polynomial iterations. Trivially we have tf \u2a7dq + 1. In fact, by the Birthday Paradox one expects that tf is of order q1/2 and there are examples of polynomials which have orbits of about this length. Here we obtain a nontrivial upper bound on the orbit length of stable quadratic polynomials: Theorem 1. For any odd q and any stable quadratic polynomial f \u2208Fq[X] we have tf = O \u0000q3/4\u0001 . By [6, Proposition 3], a quadratic polynomial f \u2208Fq[X] is stable if the adjusted orbit Orb(f) = {\u2212f(\u03b3)} [ Orb(f) contains no squares. We also recall that \u03b1 \u2208Fq is a square if either \u03b1 = 0 or \u03b1(q\u22121)/2 = 1 that can be tested (via repeated squaring) in O(log q) \ufb01eld operations. Combining these with the bound of Theorem 1, we immediately obtain: Corollary 2. For any odd q, a quadratic polynomial f \u2208Fq[X] can be tested for stability in time q3/4+o(1). Our proof is based on the Weil bound for character sums with polynomials, see [3, Theorem 11.23]. Finally, we remark that estimating the size of the set of stable quadratic polynomials aX2 + bX + c \u2208Fq[X] is a very interesting question to which we hope our technique can apply as well. 2 \f2 Proof of Theorem 1 Let \u03c7 be the quadratic character of Fq, By [6, Proposition 3], if a quadratic polynomial f \u2208Fq[X] is stable then Orb(f) contains no squares, that is, \u03c7 \u0000f (n)(\u03b3) \u0001 = \u22121, n = 2, 3, . . .. We now \ufb01x an integer parameter K and note that for any n \u2a7e1, we have simultaneously \u03c7 \u0000f (k+n)(\u03b3) \u0001 = \u22121, k = 1, . . . , K, which we rewrite as \u03c7 \u0000f (k) \u0000f (n)(\u03b3) \u0001\u0001 = \u22121, k = 1, . . . , K. (1) Since by the de\ufb01nition of tf, the values f (n)(\u03b3), n = 1, . . . , tf \u22121, are pairwise distinct elements of Fq we derive from (1) that tf \u22121 \u2a7d#Tq(K) (2) where Tq(K) = \b x \u2208Fq : \u03c7 \u0000f (k)(x) \u0001 = \u22121, k = 1, . . . , K \t . We have #Tq(K) = 1 2K X x\u2208Fq K Y k=1 \u00001 \u2212\u03c7 \u0000f (k)(x) \u0001\u0001 (3) since for every x \u2208Tq(K) the product on the right hand side of (3) is 2K; otherwise it is 0 when \u03c7(f (k)(x)) = 1 for at least one k = 1, . . . , K (note that since by our assumption f (k)(X) is irreducible over Fq we have f (k)(x) \u0338= 0 for x \u2208Fq). Just expanding the product in (3) and changing the order of summation, we obtain 2k \u22121 character sums of the shape (\u22121)\u03bd X x\u2208Fq \u03c7  \u03bd Y j=1 f (k\u03bd)(x) ! , 1 \u2a7dk1 < . . . < k\u03bd \u2a7dK, (4) with \u03bd \u2a7e1 and one trivial sum which equal to q (corresponding to the terms 1 in the product in (3)). 3 \fClearly f (k)(X) is a polynomial of degree 2k. By our assumption they are irreducible, therefore none of the polynomials \u03bd Y j=1 f (k\u03bd)(X) \u2208Fq[X], 1 \u2a7dk1 < . . . < k\u03bd \u2a7dK, is a perfect square. Therefore the Weil bound see [3, Theorem 11.23], applies to every sum (4) and implies that each of them is O(2Kq1/2). Therefore #Tq(K) = 1 2K q + O(2Kq1/2). (5) Choosing K to satisfy 2K \u2a7dq1/4 < 2K+1 and combining (2) and (5) we conclude the proof. 3 Comments It is certainly interesting to obtain nontrivial estimates on the size Sq of the set of the triples (a, b, c) \u2208F\u2217 q \u00d7Fq \u00d7Fq which correspond to stable quadratic polynomials f(X) = aX2 + bX + c. Denoting by Fk(a, b, c) the kth element of the critical orbit of f, we see that for any integer parameter K we have Sq \u2a7d#Wq(K), (6) where Wq(K) = \b (a, b, c) \u2208F\u2217 q \u00d7 Fq \u00d7 Fq : \u03c7 (Fk(a, b, c)) = \u22121, k = 1, . . . , K \t , and as before \u03c7 denotes the quadratic character of Fq. As in the proof of Theorem 1, we have #Wq(K) \u2a7d1 2K X (a,b,c)\u2208F\u2217 q\u00d7Fq\u00d7Fq K Y k=1 (1 \u2212\u03c7 (Fk(a, b, c))) (7) since for every triple (a, b, c) \u2208Wq(K) the product on the right hand side of (7) is 2K; otherwise it is either 0 (when \u03c7(Fk(a, b, c)) = 1 for at least one k = 1, . . . , K) or 1 (when F1(a, b, c) = . . . = FK(a, b, c) = 0). 4 \fClearly Fk(a, b, c) are rational functions in a, b, c of degree at most O(2k). Just expanding the product in (7) and changing the order of summation, we obtain 2k \u22121 character sums of the shape (\u22121)\u03bd X (a,b,c)\u2208F\u2217 q\u00d7Fq\u00d7Fq \u03c7  \u03bd Y j=1 Fk\u03bd(a, b, c) ! , 1 \u2a7dk1 < . . . < k\u03bd \u2a7dK, (8) with \u03bd \u2a7e1 and one trivial sum corresponding to 1 in (7). Assuming that one can prove that the Weil-type bound O(2Kq5/2) applies to all of them, we obtain from (6) that Sq = O(q3/2K + 2Kq5/2) and optimising the choice of K we derive Sq = O(q11/4). In fact, for a nontrivial estimate of Sq it is enough to show that almost all sums (8) admit a nontrivial estimate, which we pose as an open question. Acknowledgement The authors are grateful to Rafe Jones and Arne Winterhof for careful reading of the preliminary version of the manuscript and many useful comments. During the preparation of this paper, A. O. was supported in part by the Swiss National Science Foundation Grant 121874 and I. S. by the Australian Research Council Grant DP0556431."
+                }
+            ]
+        },
+        "edge_feat": {}
+    }
+}
\ No newline at end of file