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:

$\text{Mean}\;\frac{\alpha(d)}{d} = 0.1712, \qquad \text{99\% interval:}\; [0.1708,\; 0.1745]$

This is an extraordinarily tight band — relative width $\sim 2\%$. The dominant continued fraction prefix is $[0;\, 5, 1, \ldots]$ for 99.7% of all $d > 1000$.

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	99.91%
4	0.07%
$\leq 3$	0.02%

For 99.91% 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 complete table of $(d, \alpha(d))$ pairs for $d = 1$ to $100{,}000$ is available at github.com/cahlen/idontknow/data/zaremba-witnesses/. The file witnesses_1_100000.csv contains columns d, alpha, ratio, cf_prefix, max_quotient. SHA-256 checksum will be published alongside the dataset for independent verification.

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.