The {1,k} Density Hierarchy

The Finding

For each $k = 2, 3, \ldots, 10$, we computed the Zaremba density of the pair $A = \{1, k\}$ at $N = 10^{10}$. The density drops exponentially with $k$:

$k$	Density at $10^{10}$	$\dim_H(E_{\{1,k\}})$	Above $1/2$?	Ratio to $k-1$
2	76.5487%	0.531	Yes	—
3	11.0568%	0.454	No	6.9x drop
4	1.6096%	0.397	No	6.9x drop
5	0.4398%	0.349	No	3.7x drop
6	0.1721%	0.309	No	2.6x drop
7	0.0840%	0.275	No	2.0x drop
8	0.0475%	0.246	No	1.8x drop
9	0.0297%	0.221	No	1.6x drop
10	0.0201%	0.199	No	1.5x drop

Why This Matters

The critical jump is at $k = 2$

At $N = 10^{10}$, the density ratio $\rho(\{1,2\}) / \rho(\{1,3\}) = 76.55 / 11.06 \approx 6.9$. This is the largest consecutive ratio in the hierarchy. It measures the density gap between two specific digit pairs at a fixed search range, not an intrinsic “value” of digit 2 vs. digit 3; at different $N$ the ratio may shift (though we expect it to stabilize as $N \to \infty$ because both sets have positive Hausdorff dimension). The large jump reflects both $\{1,2\}$ crossing the Hausdorff dimension threshold ($\delta > 1/2$) and the Gauss measure weight $1/k^2$ dropping by a factor of $4/9 \approx 0.44$ from $k=2$ to $k=3$.

Gauss measure predicts the hierarchy

The Gauss measure assigns weight proportional to $\log(1 + 1/(a(a+2)))$ to digit $a$ in a typical continued fraction. For small $a$:

$a$	Gauss weight	Relative to $a=1$
1	0.415	1.00
2	0.170	0.41
3	0.093	0.22
4	0.059	0.14
5	0.041	0.10

Digit 1 appears 41.5% of the time in a typical CF. Digit 2 appears 17%. Digit 3 appears 9.3%. The exponential decay in our density hierarchy directly reflects this concentration: pairs with rarer digits produce exponentially fewer CF representations, leading to exponentially lower density.

Power-law fit

The densities fit approximately:

$\text{density}(\{1,k\}) \approx C \cdot k^{-\alpha} \qquad \text{for } k \geq 3$

with $\alpha \approx 5.18$ (OLS on $\log_{10}$-transformed data for $k = 3, \ldots, 10$; 95% confidence interval $[4.65,\, 5.71]$, $R^2 = 0.99$, $n = 8$). This is steeper than twice the Gauss measure exponent ($2 \times 2 = 4$), likely because the density of a restricted digit set depends not only on individual digit frequencies but on the full spectral gap of the transfer operator, which decays faster than $1/k^2$ alone would predict.

Without Digit 1: The {2,k} Hierarchy

For comparison, we computed all $\{2, k\}$ pairs at $10^{10}$:

$k$	$\{1,k\}$ density	$\{2,k\}$ density	Digit 1 multiplier
3	11.06%	0.0455%	243x
4	1.61%	0.0106%	152x
5	0.44%	0.0041%	107x
6	0.172%	0.0023%	75x
7	0.084%	0.0013%	65x
8	0.047%	0.0009%	55x
9	0.030%	0.0006%	47x
10	0.020%	0.0005%	42x

Digit 1 amplifies density by 42—243x over the equivalent pair with digit 2 (ratios computed from the same GPU kernel at $N = 10^{10}$; see results/gpu_A1k_1e10.log and results/gpu_A2k_1e10.log for raw counts). The amplification is strongest for small $k$ (where digit 1’s presence lifts the Hausdorff dimension above the critical threshold) and weakest for large $k$ (where both sets have such low dimension that density is near zero regardless).

Without digit 1, no pair achieves even 0.1% density. This is the strongest quantitative evidence for the digit 1 dominance phenomenon.

Closed Exception Sets

Four $\{1, 2, k\}$ triples have computationally observed exception sets that appear stable — no new exceptions appear when extending the search range by a factor of 10. This is observational stability, not a proof of finiteness. No branch-and-bound or analytic argument rules out further exceptions beyond our search range. The search is exhaustive within the stated range (every integer $1 \leq d \leq N$ is checked via the bitset).

Digit set	Exceptions	Exhaustive to	Stability window	Status
$\{1,2,3\}$	27	$10^{10}$	$10^9 \to 10^{10}$: no growth	$10^{11}$ in progress
$\{1,2,4\}$	64	$10^{10}$	$10^9 \to 10^{10}$: no growth	$10^{11}$ in progress
$\{1,2,5\}$	374	$10^{11}$	$10^{10} \to 10^{11}$: no growth	stable
$\{1,2,6\}$	1,834	$10^{11}$	$10^{10} \to 10^{11}$: no growth	stable

The largest exception for $\{1,2,4\}$ is $d = 51{,}270$ (full list of all 64 values available in results/gpu_A124_1e10.log).

The sequence 27, 64, 374, 1,834 grows rapidly with $k$. We cannot rigorously prove these sets are finite — additional exceptions could in principle appear beyond our search range. However, the stability across a full decade of extension is strong computational evidence. The computation A=$\{1,2,7\}$ at $10^{10}$ found 7,178 uncovered; the $10^{11}$ run is underway to test stability.

Reproduce

nvcc -O3 -arch=sm_100a -o zaremba_density_gpu scripts/experiments/zaremba-density/zaremba_density_gpu.cu -lm
for k in 2 3 4 5 6 7 8 9 10; do
    ./zaremba_density_gpu 10000000000 1,$k
done

Algorithm. The kernel enumerates all continued fractions $[a_1, a_2, \ldots]$ with $a_i \in A$ by DFS over the CF tree. Each node corresponds to a convergent $p_n/q_n$; children are formed via $q_{n+1} = a \cdot q_n + q_{n-1}$ for each $a \in A$, pruning when $q > N$. Reachable denominators are marked in a global bitset (1.25 GB for $N = 10^{10}$, one bit per integer). The CPU generates prefixes to depth 4—12 (depending on $|A|$ and $N$), then launches one GPU thread per prefix for the remaining DFS. Bit-marking uses atomicOr for thread safety. After GPU completion, the CPU counts marked bits.

Timing per pair (NVIDIA B200, CUDA 12.8, nvcc -O3 -arch=sm_100a):

Pair	GPU enum (s)	Total (s)	Prefixes
{1,2}	79.8	88.4	4096
{1,3}	9.3	18.0	4096
{1,4}	2.4	11.1	4096
{1,5}	1.8	10.4	4096
{1,6}	1.9	10.6	4096
{1,7}	1.7	10.3	4095
{1,8}	1.6	10.3	4083
{1,9}	1.5	10.3	4083
{1,10}	1.4	10.1	4017

The large tree for $\{1,2\}$ (Hausdorff dimension 0.531) takes 88 s; all other pairs complete in 10—18 s. Full output logs are in scripts/experiments/zaremba-density/results/.

---

Computed 2026-04-01 on NVIDIA B200. Human-AI collaboration (Cahlen Humphreys + Claude). Not peer-reviewed.