Digit 1 Amplification in Zaremba Density

The Finding

For Zaremba density — the fraction of integers $d \leq N$ representable as a continued fraction denominator using only digits from a set $A$ — replacing digit 2 with digit 1 amplifies density by a large factor. The original inverse-square headline is too strong: the current matched-cutoff data show strong amplification, but not a confirmed $k^{-2}$ law.

At matched $N=10^{10}$ for both $\{1,k\}$ and $\{2,k\}$:

$\frac{\text{density}(\{1,k\})}{\text{density}(\{2,k\})} \approx 1143\,k^{-1.463}\qquad (R^2=0.990,\ k=3,\ldots,10).$

At $N=10^{11}$, matched data currently exist only for $k=3,4,5$, giving a suggestive but underdetermined fit $\approx 3562\,k^{-1.93}$. That three-point overlap is not enough to claim an inverse-square law.

Data

All densities in this table use the same cutoff, $N=10^{10}$:

$k$	density $\{1,k\}$	density $\{2,k\}$	Amplification
3	11.0568%	0.045486%	243×
4	1.6096%	0.010605%	152×
5	0.4398%	0.004126%	107×
6	0.1721%	0.002333%	73.8×
7	0.0840%	0.001302%	64.6×
8	0.0475%	0.000869%	54.6×
9	0.0297%	0.000638%	46.6×
10	0.0201%	0.000472%	42.5×

The earlier version mixed $\{1,k\}$ densities at $N=10^{11}$ with $\{2,k\}$ densities at $N=10^{10}$, which biases the fitted exponent because these densities are still scale-dependent.

Why This Matters

The golden ratio connection

Digit 1 in a continued fraction corresponds to the golden ratio $\phi = [1;1,1,1,\ldots]$. The Gauss measure assigns weight $\log_2(1 + 1/(a(a+2)))$ to digit $a$, giving digit 1 approximately 41.5% of the total weight — nearly half. But Gauss measure is about typical behavior. Our result quantifies the extremal behavior: how much more of the integer line digit 1 can reach compared to digit 2, as a function of the companion digit.

Why an inverse-square law is still plausible but unproved

The Gauss measure weight ratio between digits 1 and 2 is:

$\frac{\log(1 + 1/3)}{\log(1 + 1/8)} = \frac{\log(4/3)}{\log(9/8)} \approx 2.41$

This is a constant — it doesn’t depend on $k$. So the $1/k^2$ decay in amplification must come from the interaction between digit 1 and digit $k$, not from digit 1 alone. As $k$ grows, the continued fraction tree for $\{1,k\}$ and $\{2,k\}$ become increasingly similar in structure (both dominated by their small digit), and the advantage of digit 1 over digit 2 diminishes — but at a rate governed by $1/k^2$.

This is reminiscent of the $\sum 1/a^2$ predictor for Hausdorff dimension (see Hausdorff digit-one dominance): the difference $1/1^2 - 1/2^2 = 3/4$ is a constant, but its relative importance compared to $1/k^2$ decays as $k$ grows. The current data are consistent with a drift toward a quadratic law at larger $N$, but matched larger-scale runs are required before making that claim.

Additional Patterns

{1,2,k} exception count growth

The number of integers with no $\{1,2,k\}$-representation grows as a decelerating exponential in $k$:

$k$	Exceptions	Ratio $E(k)/E(k-1)$	Status
3	27	—	Stable candidate through $10^{10}$; 10^11 repo log is partial
4	64	2.4	Stable candidate through $10^{10}$; 10^11 repo log is partial
5	374	5.8	Stable candidate through $10^{10}$; 10^11 repo log is partial
6	1,834	4.9	Stable candidate through $10^{11}$
7	7,178	3.9	Stable candidate through $10^{11}$
8	23,590	3.3	Open at $10^{11}$
9	77,109	3.3	Open at $10^{11}$
10	228,514	3.0	Open at $10^{11}$

The growth ratios converge toward $\approx 3.2$, suggesting asymptotic behavior $E(k) \sim C \cdot 3.2^k$. A quadratic-log model $\log E(k) = -0.030k^2 + 1.73k - 1.90$ fits with $R^2 = 0.997$.

{1,3,5} approaching a finite limit

The exception set for $A = \{1,3,5\}$ shows strong evidence of convergence:

$N$	Exceptions	New exceptions
$10^9$	75,547	—
$10^{10}$	80,431	+4,884
$10^{11}$	80,945	+514

The increment dropped 9.5× per decade. Geometric extrapolation gives a limit of approximately 81,005, with fewer than 60 exceptions remaining beyond $10^{11}$. If confirmed at $10^{12}$, this would be evidence for another stable exception set and the first such candidate with non-consecutive digits.

Method

• Transfer operator enumeration: GPU threads enumerate all continued fraction trees with digits in $A$, marking denominators in a bitset
• Prefix splitting: CPU generates tree prefixes to bounded depth; GPU threads do DFS on subtrees
• Hardware: 8×NVIDIA B200 GPUs (180 GB each), RTX 5090 for smaller runs
• All densities computed from complete enumeration up to $N$, not sampling

Connection to Other Findings

• Zaremba digit pair hierarchy: Established the $\{1,k\}$ and $\{2,k\}$ hierarchies separately; this finding quantifies the ratio between them
• Zaremba exception hierarchy: Exception counts for $\{1,2,k\}$; this finding adds the growth model and {1,3,5} convergence
• Hausdorff digit-one dominance: The $\sum 1/a^2$ predictor for dimension motivates testing inverse-square amplification in density, but does not prove it

Code

• Density kernel: scripts/experiments/zaremba-density/zaremba_density_gpu.cu
• All results: scripts/experiments/zaremba-density/results/

References

• Zaremba, S. K. (1972). “La méthode des ‘bons treillis’ pour le calcul des intégrales multiples.” In 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.
• Hensley, D. (1996). “A polynomial time algorithm for the Hausdorff dimension of continued fraction Cantor sets.” Journal of Number Theory, 58(1), pp. 9–45.

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Computed on 8×NVIDIA B200 GPUs. All data open at github.com/cahlen/idontknow.

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