Zaremba Witnesses Concentrate at $\alpha(d)/d \approx 0.171$

The Finding

For the smallest Zaremba witness $\alpha(d)$ with bound $A = 5$, the ratio $\alpha(d)/d$ is remarkably concentrated:

$\text{Mean}\;\frac{\alpha(d)}{d} = 0.171232, \qquad \text{99\% interval:}\; [0.1708,\; 0.1751]$

This is an extraordinarily tight band — relative width $\sim 3\%$. The dominant continued fraction prefix is $[0;\, 5, 1, \ldots]$ for 99.72% of all $d \le 10^5$, rising to 99.93% for $d > 1000$.

Revision (2026-07-21). The raw witness table backing this page was regenerated from scratch and committed (see Data Availability below); the 2026-07-21 audit had flagged its absence. The regeneration reproduces the original headline claims (mean 0.171232 vs. claimed 0.1712; median 0.1709; 1st percentile 0.1708), with two corrections: the 99th percentile is 0.1751 (previously stated 0.1745), and the 99.7% prefix figure applies to all $d \le 10^5$ (for $d > 1000$ the correct figure is 99.93%). The max-quotient tightness table was recomputed under an explicit convention. Witness minimality was verified by brute force on 45 sampled denominators with zero failures.

The Golden Ratio Connection

The infinite continued fraction $[0;\, 5, 1, 1, 1, \ldots]$ converges to:

$\frac{1}{5 + \varphi} = \frac{1}{5 + \frac{1+\sqrt{5}}{2}} \approx 0.1511$

where $\varphi$ is the golden ratio. The observed mean $0.1712$ lies between the finite convergents:

$[0;\, 5, 1] = \frac{1}{6} \approx 0.1667 \qquad \text{and} \qquad [0;\, 5, 1, 1] = \frac{2}{11} \approx 0.1818$

The witnesses cluster in this window because the optimal CF starts with the maximum allowed quotient (5), then drops to the minimum (1), then varies — balancing the constraint of keeping all quotients $\leq 5$ while being coprime to $d$.

Caveat on the golden ratio. The infinite CF $[0; 5, 1, 1, 1, \ldots]$ converges to $1/(5+\varphi) \approx 0.1511$, which is 13% below the observed mean of $0.1712$. The witnesses are finite (typically 3–6 terms) and cluster between the low-order convergents $1/6 \approx 0.1667$ and $2/11 \approx 0.1818$, not near the infinite limit. The appearance of the golden ratio is a heuristic observation about why digit 1 dominates after the leading 5 (digit 1 is the greedy choice that maximizes remaining CF flexibility), not a proven structural connection. A rigorous explanation for the concentration at $0.171$ remains open.

Tightness of $A = 5$

Max quotient used	Frequency
5	99.934%
4	0.049%
$\leq 3$	0.017%

(Convention: each rational has two CF expansions; “max quotient used” is the smaller of the two maxima among expansions achieving the minimal witness.) For 99.93% of moduli $d \leq 100{,}000$, the smallest witness $\alpha(d)$ requires a partial quotient of exactly 5. This does not imply that no witness with all quotients $\leq 4$ exists for those $d$ — larger witnesses may satisfy a tighter bound. The statistic measures minimal-witness tightness, not global tightness of Zaremba’s conjecture at $A = 4$.

Practical Impact

This observation enabled a 13x speedup in our CUDA verification kernel (v2 over v1) by starting the witness search at $a = \lfloor 0.170d \rfloor$ instead of $a = 1$. Note: v2 also incorporated other optimizations (loop unrolling, early-exit predicates), so the full 13x factor is not solely attributable to the starting offset.

Code

Analysis script and raw data: github.com/cahlen/idontknow

Data Availability

The full witness table is committed: witnesses_100000.csv (columns d, alpha, ratio, a1, a2, max_digit; SHA-256 414dd177e6fe963a3d62fa3af4feafb9289ccad1a044058cfc333447612567a3), generated by witness_minimal.c — a single DFS over admissible CF words tracking both convergent continuants (0.7 s, 242M nodes; run log run_witness_minimal_1e5.log). Consecutive convergents are automatically coprime, so every row is a valid witness by construction; minimality was additionally brute-force checked on a 45-denominator sample.

References

1. Zaremba, S.K. (1972). “La méthode des ‘bons treillis’ pour le calcul des intégrales multiples.” Applications of Number Theory to Numerical Analysis, pp. 39–119.
2. Bourgain, J. and Kontorovich, A. (2014). “On Zaremba’s conjecture.” Annals of Mathematics, 180(1), pp. 137–196. arXiv:1107.3776
3. Khintchine, A.Ya. (1964). Continued Fractions. University of Chicago Press.

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Computed from exhaustive analysis of $d = 1$ to $100{,}000$ on NVIDIA DGX B200.

This work was produced through human–AI collaboration (Cahlen Humphreys + Claude). Not independently peer-reviewed. All code and data open for verification at github.com/cahlen/idontknow.