cahlen/continued-fraction-spectra

Continued Fraction Spectra

Five GPU-computed datasets exploring the geometry and dynamics of continued fractions. AI-audited, not peer-reviewed.

Part of the bigcompute.science project.

Datasets

1. Hausdorff Dimension Spectrum (hausdorff-spectrum/)

dim_H(E_A) for continued fraction Cantor sets, computed for all nonempty subsets of {1,...,n}.

File	Subsets	Description
spectrum_n20.csv	1,048,575	All subsets of {1,...,20} — complete
spectrum_n10.csv	1,024	All subsets of {1,...,10}
spectrum_n5.csv	32	All subsets of {1,...,5}

Columns: subset_mask, subset_digits, cardinality, max_digit_in_subset, dimension

Validated against:

• dim_H(E_{1,...,5}) = 0.836829443681208 (Jenkinson-Pollicott)
• dim_H(E_{1,2}) = 0.531280506277205 (known value)

Key finding — Digit 1 dominance: dim_H(E_{1,...,5}) = 0.837 > dim_H(E_{2,...,20}) = 0.768. Five digits with 1 produce a larger Cantor set than nineteen digits without it. See finding.

Method: Transfer operator eigenvalue via bisection (55 steps) + Chebyshev collocation (order 40). 4,343 seconds on RTX 5090.

2. Lyapunov Exponent Spectrum (lyapunov-spectrum/)

Lyapunov exponents for all nonempty subsets.

File	Subsets	Description
spectrum_n20.csv	1,048,575	All subsets of {1,...,20} — complete
spectrum_n10.csv	1,024	All subsets of {1,...,10}
spectrum_n5.csv	32	All subsets of {1,...,5}

Columns: subset_mask, subset_digits, cardinality, spectral_radius_s1, lyapunov_exponent

Method: Transfer operator with Chebyshev collocation (order 40), 300 power iterations. 305 seconds on RTX 5090.

3. Minkowski ?(x) Singularity Spectrum (minkowski-spectrum/)

Multifractal singularity spectrum f(alpha) of the Minkowski question-mark function.

File	Records	Description
spectrum.csv	2,001	f(alpha) for q in [-10, 10], step 0.01

Columns: q, tau_q, alpha_q, f_alpha

Note: First ~200 rows have NaN (q < -8 diverges). Meaningful range: q in [-8, 10]. f_alpha_max = 0.987, alpha support = [0.747, 4.459]. 4.9 seconds on RTX 5090.

4. Flint Hills Series (flint-hills/)

Partial sums of sum(1/(n^3 * sin(n)^2)) computed to N = 10^10.

File	Records	Description
partial_sums.csv	5	S_N at N = 10^6 through 10^10
growth_rate.csv	19	Growth rate per decade
spikes.csv	20	Top 20 spike contributions

Key finding: Spikes (terms near rational approximants of pi) contribute 91.2% of the total sum. Used quad-double precision (~62 digits) for spike calculations.

5. Prime Convergents (prime-convergents/)

Primality statistics for convergent numerators and denominators. Extends Humphreys (2013, NCUR/Boise State).

File	Samples	Description
stats_random_10M_d500.csv	10,000,000	Random CFs (Gauss-Kuzmin), depth 500
metadata_random_10M_d500.json	1	Run metadata and aggregate statistics

Columns: sample_id, depth, prime_An, prime_Bn, doubly_prime, mean_ratio, min_ratio, overflow_depth

Key findings:

• Erdos-Mahler bound G(A_n) >= e^{n/(50 ln n)} holds for all 10M samples (100%)
• The constant 50 is very conservative: worst-case ratio is 4.87, mean is 116.7
• Average 4.92 prime numerators per CF (out of ~38 terms before uint64 overflow)
• 0.95 doubly-prime convergents per CF on average; max observed is 7 in a single CF
• For e specifically: exactly 3 doubly-prime convergents (matches Humphreys 2013)

Method: GPU-parallel convergent recurrence (128-bit arithmetic) + deterministic Miller-Rabin (12 witnesses). 10M samples in 4.8 seconds on B200.

Hardware

Computation	Hardware	Time
Hausdorff n=20	RTX 5090 (32 GB)	4,343 sec
Lyapunov n=20	RTX 5090 (32 GB)	305 sec
Minkowski spectrum	RTX 5090 (32 GB)	4.9 sec
Flint Hills 10^10	8x B200 DGX	~2 hours
Prime convergents 10M	1x B200	4.8 sec

Source

• Code: github.com/cahlen/idontknow (scripts/experiments/)
• Findings: Digit 1 dominance · Flint Hills
• MCP Server: mcp.bigcompute.science
• AGENTS.md: Contribution guide (22 tools, no auth)

Understanding This Data

When you restrict which numbers can appear in a continued fraction, the resulting set of real numbers forms a fractal. This dataset measures how "thick" each of those fractals is, for every possible restriction using digits 1 through 20.

The Hausdorff dimension file (spectrum_n20.csv) has 1,048,575 rows -- one for every non-empty subset of {1,...,20}. Each row gives a digit set and its dimension, a number between 0 and 1 where bigger means the fractal fills more of the number line. For instance, allowing digits {1,2,3,4,5} gives dimension 0.837, which is exactly the number that appears in Zaremba's conjecture.

Here is the most striking pattern in this data: the set {1,...,5} (five digits including 1) has dimension 0.837, but {2,...,20} (nineteen digits without 1) only reaches 0.768. Five digits with 1 beat nineteen digits without it. Digit 1 is disproportionately important because it produces the "widest" intervals in the continued fraction construction, so it contributes more to the fractal's thickness than any other digit.

The Lyapunov exponent files measure something related: how fast nearby orbits separate under the continued fraction map, restricted to each digit set. The Minkowski question-mark function data captures the multifractal structure of a famous function that connects continued fractions to binary expansions. The Flint Hills data tracks partial sums of a series whose convergence is still an open question.

This is the first complete enumeration of these dimensions for all subsets up to size 20. No table like this exists in the published literature.

Citation

@dataset{humphreys2026cfspectra,
  title = {Continued Fraction Spectra: Hausdorff, Lyapunov, Minkowski, Flint Hills, and Prime Convergents},
  author = {Humphreys, Cahlen},
  year = {2026},
  publisher = {Hugging Face},
  url = {https://huggingface.co/datasets/cahlen/continued-fraction-spectra}
}

Human-AI collaborative work. AI-audited against published literature. Not independently peer-reviewed. CC BY 4.0.