Exploratory BFS Depths for Zaremba Generators

The Finding

For each prime $p \leq 1021$, we computed exploratory BFS depths for the Zaremba generators. The original writeup described these as Cayley graph diameters of $\Gamma_{\{1,\ldots,5\}}$ in $\text{SL}_2(\mathbb{Z}/p\mathbb{Z})$, but the audit found that this is not yet a certified statement: the implemented generators have determinant $-1$, while $\text{SL}_2$ contains determinant $1$ matrices only.

$\frac{\text{diam}(p)}{\log p} \in [1.37, \, 3.11] \qquad \text{for all } 172 \text{ primes } p \leq 1021$

Note: 172 is the exact prime count ($\pi(1021) = 172$). Only prime moduli are tested since $\text{SL}_2(\mathbb{Z}/p\mathbb{Z})$ is well-defined as a group for prime $p$. The maximum ratio 3.11 occurs at $p = 5$; for $p \geq 7$ the ratio stays below 2.89.

The ratio is decreasing and appears to converge to approximately $1.45$, suggesting:

$\text{diam}(p) \leq 2 \log p \qquad \text{for all sufficiently large } p$

This asymptotic claim is based on the empirical trend over two orders of magnitude in $p$ and should be treated as a conjecture, not a proven bound. Quantitatively: over the range $p \in [101, 1021]$ (108 primes), the reported ratio drops toward 1.44 at $p=1021$. A corrected computation must first certify the ambient group, unique visited count, and no frontier clipping before these values can be cited as Cayley graph diameters.

The maximum diameter observed is 10, first achieved at $p = 211$.

Why This Matters

Direct Bound on Continued Fraction Length

The Cayley diameter equals the maximum continued fraction length needed to represent any element of $\text{SL}_2(\mathbb{Z}/p\mathbb{Z})$ as a product of the generators $g_a = \begin{pmatrix} a & 1 \\ 1 & 0 \end{pmatrix}$. Since continued fraction convergents correspond to products of these matrices, the diameter bounds the worst-case CF length needed to hit any denominator class modulo $p$.

Note: the correspondence between Cayley diameter in $\text{SL}_2(\mathbb{Z}/p\mathbb{Z})$ and CF length over $\mathbb{Z}$ is subtle — reduction mod $p$ can lose information. The diameter gives an upper bound on CF length modulo $p$, but lifting to integer continued fractions requires additional arguments (see Bourgain-Kontorovich 2014, Section 4).

If $\text{diam}(p) \leq C \log p$ with explicit $C$, this could feed into an effective $Q_0$ for Zaremba’s Conjecture via the Bourgain-Kontorovich circle method (Bourgain-Kontorovich, 2014).

Connection to Property (τ)

The logarithmic diameter growth is consistent with property (τ) — the spectral gap of the Cayley graph Laplacian remains bounded away from zero as $p \to \infty$. Combined with our spectral gap data ($\sigma_m \geq 0.237$ for all square-free $m \leq 1999$) and transitivity proof, this provides three independent lines of evidence that the semigroup has strong expansion properties.

The connection between spectral gap and diameter is classical: for a $k$-regular Cayley graph on $n$ vertices with spectral gap $\lambda_1$, the diameter satisfies $\text{diam} \leq \lceil \log n / \log(k/\lambda_1) \rceil$ (Chung, 1989). Our spectral gap data predicts exactly the logarithmic diameter growth we observe.

Comparison with Known Results

Helfgott’s growth theorem (Helfgott, 2008) implies that generating sets of $\text{SL}_2(\mathbb{F}_p)$ produce Cayley graphs with diameter $O(\log p)$, but without explicit constants. Bourgain-Gamburd (Bourgain-Gamburd, 2008) proved that random generators give spectral gaps bounded away from zero, yielding $O(\log p)$ diameter. Because the current Zaremba-generator kernel mixes determinant $-1$ generators with an $\text{SL}_2$ group-size target, these runs should be treated as exploratory numerical data for the generator dynamics, not certified $\text{SL}_2$ Cayley diameters.

Data

Prime range	Count	Max diam	Max ratio	Min ratio
$p \leq 50$	15	5	2.89	1.44
$50 < p \leq 100$	10	7	2.52	1.53
$100 < p \leq 200$	21	8	1.79	1.51
$200 < p \leq 500$	62	10	1.61	1.45
$500 < p \leq 1021$	87	10	1.44	1.37

Sample data points:

$p$	$\|\text{SL}_2\|$	diam	$\log p$	diam/$\log p$
2	6	2	0.69	2.89
5	120	5	1.61	3.11
13	2184	6	2.56	2.34
101	1030200	8	4.62	1.73
211	9393480	10	5.35	1.87
509	131906220	10	6.23	1.61
1021	1064296620	10	6.93	1.44

Method

For each prime $p$:

1. Encode elements of $\text{SL}_2(\mathbb{Z}/p\mathbb{Z})$ as integers: $\begin{pmatrix} a & b \\ c & d \end{pmatrix} \mapsto a \cdot p^3 + b \cdot p^2 + c \cdot p + d$
2. GPU BFS from the identity $I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$, expanding all frontier nodes in parallel
3. Each thread computes 10 neighbors (5 generators + 5 inverses) via right-multiplication
4. Visited set: bitset of size $p^4/8$ bytes with atomicOr for lock-free marking
5. Frontier double-buffered: current → next, swap each level
6. Diameter = number of BFS levels until frontier is empty
7. Required validation for certification: after BFS completes, count set bits in the visited bitset and verify it equals the order of the intended ambient group; separately assert that the frontier buffer never clipped. The current kernel does not yet provide this full certificate.

Measured throughput for $p = 1021$: $\sim\!1.06 \times 10^9$ nodes visited in $\sim\!5$ seconds on 8× B200 GPUs, yielding $\sim\!2.1 \times 10^8$ BFS node expansions per second per GPU. Total wall time across all 172 primes was 40 seconds.

The group $\text{SL}_2(\mathbb{Z}/p\mathbb{Z})$ has order $p(p^2 - 1)$. For $p = 1021$: $|\text{SL}_2| \approx 1.06 \times 10^9$, bitset $\approx 130$ MB.

References

• 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.
• Bourgain, J. and Kontorovich, A. (2014). “On Zaremba’s conjecture.” Annals of Mathematics, 180(1), pp. 137–196. arXiv:1107.3776
• Bourgain, J. and Gamburd, A. (2008). “Uniform expansion bounds for Cayley graphs of $\text{SL}_2(\mathbb{F}_p)$.” Annals of Mathematics, 167(2), pp. 625–642.
• Helfgott, H.A. (2008). “Growth and generation in $\text{SL}_2(\mathbb{Z}/p\mathbb{Z})$.” Annals of Mathematics, 167(2), pp. 601–623. arXiv:math/0509024
• Chung, F.R.K. (1989). “Diameters and eigenvalues.” Journal of the AMS, 2(2), pp. 187–196.
• Dickson, L.E. (1901). Linear Groups with an Exposition of the Galois Field Theory. B.G. Teubner, Leipzig.

Code

# Compile (requires CUDA)
nvcc -O3 -arch=sm_100a -o cayley_gpu scripts/experiments/zaremba-cayley-diameter/cayley_gpu.cu

# Run for all primes up to 1021
./cayley_gpu 1021

Source: scripts/experiments/zaremba-cayley-diameter/cayley_gpu.cu

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Computed on 8× NVIDIA B200 (1.43 TB VRAM, Blackwell architecture, CUDA 12.8), 40 seconds total. Full BFS logs and per-prime diameter data available in the GitHub repository.

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.