A={1,2} Density Grows Logarithmically, Not Power-Law

The Finding

For the digit set $A = \{1,2\}$ (Hausdorff dimension $\delta = 0.531$, barely above the critical threshold $1/2$), the Zaremba density as a function of range $N$ fits a logarithmic model almost exactly:

$\text{density}(N) \approx 30.1 + 4.65 \cdot \log_{10}(N)$

Range $N$	Observed density	Predicted (log model)	Residual
$10^6$	57.98%	57.96%	-0.02%
$10^9$	72.06%	71.97%	+0.09%
$10^{10}$	76.55%	76.62%	-0.07%
$10^{12}$	84.58%	85.88%	-1.30%

The logarithmic model predicted 85.9% at $10^{12}$; the observed value is 84.58% — off by 1.3 percentage points. The pure power-law model predicted ~80%, which is further off. The truth is between the two models, but the logarithmic fit is degrading. With four data points, a revised fit gives:

$\text{density}(N) \approx 27.0 + 4.80 \cdot \log_{10}(N) \quad (\text{residuals} \leq 0.6\%)$

Updated Predictions

Range	Original prediction	Revised prediction
$10^{13}$	90.5%	89.4%
$10^{15}$	99.9%	99.0%
100% at	$10^{15.0}$	$10^{15.2}$

The convergence is slightly slower than the 3-point model suggested, but still logarithmic. Full density around $10^{15}$ remains plausible.

Why This Matters

Relationship to BK Framework

The Bourgain-Kontorovich transfer operator framework predicts the representation count grows as $R(d) \sim d^{2\delta - 1}$. For $A = \{1,2\}$:

$2\delta - 1 = 2(0.531) - 1 = 0.062$

Important distinction: the exponent 0.062 describes the growth of $R(d)$ (how many CF representations each $d$ has), not the rate at which density (the fraction of $d$ with $R(d) \geq 1$) converges to 100%. Density convergence depends on the full distribution of $R(d)$ across integers, not just its mean growth. These are related but different quantities.

Our density data fits a logarithmic model, but with only three data points this cannot definitively distinguish logarithmic from power-law convergence. The models diverge at $10^{12}$: logarithmic predicts ~86%, while a power-law fit predicts ~80%. This is a testable prediction.

What this could mean

1. Pre-asymptotic regime: At $N \leq 10^{10}$, the system hasn’t yet reached the true asymptotic behavior. The logarithmic fit may break down at $N > 10^{12}$ and transition to the slower power-law predicted by BK.

2. Corrections to the leading term: The BK counting formula has error terms. If the error term is $O(N^{2\delta - 1 - \varepsilon})$ with $\varepsilon$ small, the effective growth rate could appear faster than the leading exponent suggests.

3. Logarithmic corrections: Some number-theoretic counting functions have $\log(N)$ corrections. If $R(d) \sim d^{2\delta-1} \cdot (\log d)^c$ for some $c > 0$, this could produce the observed logarithmic density growth.

Testable prediction

The model makes a sharp prediction: density at $10^{12}$ should be ~85.9%. If the observed density at $10^{12}$ is significantly different (e.g., 82% or 89%), it would distinguish between logarithmic and power-law convergence.

Computing $A = \{1,2\}$ at $10^{12}$ requires ~100× more work than $10^{10}$ (about 10 hours on B200). This is a feasible next experiment.

The Digit 1 Advantage: A Sigmoid

Our complete density sweep of all 1,023 subsets of $\{1, \ldots, 10\}$ reveals that digit 1’s advantage follows a sigmoid that peaks at cardinality 4:

Cardinality	Avg density (with 1)	Avg density (without 1)	Gap
2	11.0%	0.1%	10.9 pp
3	58.9%	1.8%	57.1 pp
4	92.1%	12.7%	79.4 pp
5	99.5%	39.4%	60.0 pp
6	100.0%	70.9%	29.1 pp
7	100.0%	91.8%	8.2 pp
8	100.0%	99.2%	0.8 pp

At cardinality 4, digit 1 is worth 79 percentage points of density. By cardinality 8, the advantage shrinks to under 1 point — enough other digits compensate.

Exception Scaling: {1,2,k} Follows k^7

For the family $A = \{1, 2, k\}$ at $N = 10^6$, the number of uncovered integers grows approximately as $k^7$:

$k$	Exceptions	Ratio to $k-1$
3	27	—
4	64	2.4
5	373	5.8
6	1,720	4.6
7	5,388	3.1
8	11,746	2.2
9	21,796	1.9
10	33,025	1.5

Adding larger third digits helps rapidly less. The “sweet spot” is $k = 3$ (27 exceptions) — adding digit 4 gives 64 exceptions (2.4×), but adding digit 10 gives 33,025 (1,223×).

Reproduce

git clone https://github.com/cahlen/idontknow
cd idontknow

# GPU computation
nvcc -O3 -arch=sm_100a -o zaremba_density_gpu scripts/experiments/zaremba-density/zaremba_density_gpu.cu -lm
./zaremba_density_gpu 10000000000 1,2     # A={1,2} at 10^10
./zaremba_density_gpu 1000000 1,2,3       # A={1,2,3} at 10^6

References

1. Bourgain, J. and Kontorovich, A. (2014). “On Zaremba’s conjecture.” Annals of Mathematics, 180(1), pp. 137–196.
2. Hensley, D. (1996). “A polynomial time algorithm for the Hausdorff dimension of continued fraction Cantor sets.” J. Number Theory, 58(1), pp. 9–45.
3. Jenkinson, O. and Pollicott, M. (2001). “Computing the dimension of dynamically defined sets.” Ergodic Theory Dynam. Systems, 21(5), pp. 1429–1445.

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Computed 2026-04-01 on 8× NVIDIA B200. This work was produced through human–AI collaboration (Cahlen Humphreys + Claude). Not independently peer-reviewed. All code and data open for verification.

Caveat: Three data points make a fit, not a proof. The logarithmic model needs confirmation at $10^{12}$ and beyond. The prediction of 100% at $10^{15}$ is speculative until more data points are collected.