| import OAI.NumberTheory.TwoPoint.Fourier.MinorArcPacking |
| import Mathlib.NumberTheory.Harmonic.Bounds |
|
|
| namespace OAI |
|
|
| /-! Summing a truncated reciprocal over separated real representatives. -/ |
|
|
| namespace TwoPointCorrelations |
|
|
| open Finset |
| open scoped Classical |
|
|
| theorem minor_arc_packing_sum {ι : Type*} (S : Finset ι) (x w : ι → ℝ) |
| (δ V R : ℝ) (hδ : 0 < δ) (hV : 0 ≤ V) |
| (hsep : ∀ i ∈ S, ∀ j ∈ S, i ≠ j → δ ≤ |x i - x j|) |
| (hx : ∀ i ∈ S, |x i| ≤ R) (hw : ∀ i ∈ S, w i ≤ V) |
| (hwinv : ∀ i ∈ S, 0 < |x i| → w i ≤ 1 / |x i|) : |
| (∑ i ∈ S, w i) ≤ 2 * V + (2 / δ) * (1 + Real.log (⌈R / δ⌉₊ : ℝ)) := by |
| let L := ⌈R / δ⌉₊ |
| let b := fun i => ⌊|x i| / δ⌋₊ |
| let F := fun j : ℕ => S.filter (fun i => b i = j) |
| have hindex (i : ι) (hi : i ∈ S) : b i ≤ L := by |
| apply Nat.floor_le_of_le |
| exact (div_le_div_of_nonneg_right (hx i hi) hδ.le).trans (Nat.le_ceil _) |
| have hband (j : ℕ) (i : ι) (hi : i ∈ F j) : |
| (j : ℝ) * δ ≤ |x i| ∧ |x i| < (j : ℝ) * δ + δ := by |
| have he : b i = j := (mem_filter.mp hi).2 |
| have hl := Nat.floor_le (div_nonneg (abs_nonneg (x i)) hδ.le) |
| have hu := Nat.lt_floor_add_one (|x i| / δ) |
| change ((b i : ℕ) : ℝ) ≤ |x i| / δ at hl |
| change |x i| / δ < (b i : ℝ) + 1 at hu |
| rw [he] at hl hu |
| constructor |
| · exact (le_div_iff₀ hδ).mp hl |
| · have hh := (div_lt_iff₀ hδ).mp hu |
| nlinarith |
| have hcard (j : ℕ) : (F j).card ≤ 2 := by |
| refine (card_le_card (s := F j) (t := S.filter (fun i => |
| (j : ℝ) * δ ≤ |x i| ∧ |x i| < (j : ℝ) * δ + δ)) ?_).trans |
| (minor_arc_absolute_band_packing S x δ ((j : ℝ) * δ) hsep) |
| intro i hi |
| exact mem_filter.mpr ⟨(mem_filter.mp hi).1, hband j i hi⟩ |
| have hzero : (∑ i ∈ F 0, w i) ≤ 2 * V := by |
| calc |
| _ ≤ ∑ _i ∈ F 0, V := sum_le_sum (fun i hi => hw i (mem_filter.mp hi).1) |
| _ = ((F 0).card : ℝ) * V := by simp |
| _ ≤ 2 * V := mul_le_mul_of_nonneg_right (by exact_mod_cast hcard 0) hV |
| have hpiece (j : ℕ) (hj : j ∈ Icc 1 L) : (∑ i ∈ F j, w i) ≤ 2 / (δ * j) := by |
| have hj0 : (0 : ℝ) < j := by exact_mod_cast (mem_Icc.mp hj).1 |
| have hbound (i : ι) (hi : i ∈ F j) : w i ≤ 1 / (δ * j) := by |
| have hlo := (hband j i hi).1 |
| have hpos : 0 < |x i| := (mul_pos hj0 hδ).trans_le hlo |
| apply (hwinv i (mem_filter.mp hi).1 hpos).trans |
| exact one_div_le_one_div_of_le (mul_pos hδ hj0) (by nlinarith) |
| calc |
| _ ≤ ∑ _i ∈ F j, 1 / (δ * j) := sum_le_sum hbound |
| _ = ((F j).card : ℝ) * (1 / (δ * j)) := by simp |
| _ ≤ 2 * (1 / (δ * j)) := |
| mul_le_mul_of_nonneg_right (by exact_mod_cast hcard j) (by positivity) |
| _ = _ := by ring |
| have hpartition : (∑ i ∈ S, w i) = ∑ j ∈ Icc 0 L, ∑ i ∈ F j, w i := by |
| symm |
| exact sum_fiberwise_of_maps_to (fun i hi => mem_Icc.mpr ⟨Nat.zero_le _, hindex i hi⟩) _ |
| have hsplit : (∑ j ∈ Icc 0 L, ∑ i ∈ F j, w i) = |
| (∑ i ∈ F 0, w i) + ∑ j ∈ Icc 1 L, ∑ i ∈ F j, w i := by |
| have he : Icc 0 L = insert 0 (Icc 1 L) := by |
| ext j |
| simp only [mem_Icc, mem_insert] |
| omega |
| rw [he, sum_insert (by simp)] |
| have hharm : (∑ j ∈ Icc 1 L, 2 / (δ * j)) ≤ |
| (2 / δ) * (1 + Real.log (L : ℝ)) := by |
| calc |
| _ = (2 / δ) * (harmonic L : ℝ) := by |
| rw [harmonic_eq_sum_Icc] |
| push_cast |
| rw [mul_sum] |
| apply sum_congr rfl |
| intro j _ |
| ring |
| _ ≤ _ := mul_le_mul_of_nonneg_left (harmonic_le_one_add_log L) (by positivity) |
| rw [hpartition, hsplit] |
| exact add_le_add hzero ((sum_le_sum hpiece).trans hharm) |
|
|
| end TwoPointCorrelations |
|
|
| end OAI |
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