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 $N = 10^{10}$ and $10^{11}$. The density drops exponentially with $k$:

$k$	Density at $10^{10}$	Density at $10^{11}$	$\dim_H(E_{\{1,k\}})$	Above $1/2$?
2	76.5487%	80.7543%	0.531	Yes
3	11.0568%	9.1109%	0.454	No
4	1.6096%	1.0735%	0.397	No
5	0.4398%	0.2564%	0.349	No
6	0.1721%	0.0876%	0.309	No
7	0.0840%	0.0387%	0.275	No
8	0.0475%	0.0202%	0.246	No
9	0.0297%	0.0117%	0.221	No
10	0.0201%	0.0074%	0.199	No

Why This Matters

{1,2} is the only pair whose density grows

The 10^11 data reveals something you cannot see at a single scale: $\{1,2\}$ density increases from 76.5% to 80.8% as $N$ grows from $10^{10}$ to $10^{11}$, while every other pair’s density decreases. The set $\{1,3\}$ drops from 11.1% to 9.1%. The set $\{1,10\}$ drops from 0.020% to 0.007%.

This is the Hausdorff dimension threshold at work. The dimension $\delta$ of the underlying Cantor set controls the long-term behavior: when $2\delta > 1$ (equivalently $\delta > 1/2$), the set of representable denominators is dense enough that its density converges toward 100%. When $2\delta < 1$, the set is too thin and density converges to 0%.

Only $\{1,2\}$ has $\delta = 0.531 > 1/2$. Every other pair has $\delta < 1/2$. So the 10^11 data is the first scale where we see the density trajectories clearly diverging — one pair headed toward full coverage, the rest headed toward nothing.

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 $\alpha$ in the range 3—4 (a formal log-log regression with confidence intervals has not been performed; the exponent estimate is approximate given the small number of data points). The rough magnitude is consistent with twice the Gauss measure exponent $-2$ (from $1/k^2$), which is expected since density depends on the product of the two digits’ contributions.

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	Closed
$\{1,2,6\}$	1,834	$10^{11}$	$10^{10} \to 10^{11}$: no growth	Closed
$\{1,2,7\}$	7,178	$10^{11}$	$10^{10} \to 10^{11}$: no growth	NEW — Closed

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, 7,178 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.

Update (2026-04-05): $A=\{1,2,7\}$ at $10^{11}$ confirms exactly 7,178 exceptions — unchanged from $10^{10}$, making this the fifth closed exception set. Meanwhile $\{1,2,8\}$ has 23,590 at $10^{11}$ (growing), suggesting a sharp closed/open threshold at $k=7$.

Open Exception Sets at $10^{11}$

Digit set	Exceptions	Growth from $10^{10}$	Status
$\{1,2,8\}$	23,590	growing	Open
$\{1,2,9\}$	77,109	growing	Open
$\{1,2,10\}$	228,514	growing	Open
$\{1,3,5\}$	80,945	+514 from 80,431	Slowly growing

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/.

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