Digit 1 Dominance: Five Digits With 1 Beat Fourteen Digits Without

The Finding

For the restricted continued fraction Cantor sets $E_A = \{x \in [0,1] : \text{all partial quotients of } x \text{ lie in } A\}$, the Hausdorff dimension spectrum at $n = 15$ reveals an extreme asymmetry:

$\dim_H(E_{\{1,\ldots,5\}}) = 0.837 \quad > \quad \dim_H(E_{\{2,\ldots,15\}}) = 0.747$

Five digits containing digit 1 produce a larger Cantor set than fourteen digits without it. Digit 1 alone is worth more than digits 6 through 15 combined.

Why This Matters

The Gauss measure assigns weight proportional to $\log(1 + 1/(a(a+2)))$ to digit $a$, which for small $a$ concentrates dramatically on $a = 1$. This is well-known qualitatively — the continued fraction expansion of a typical real number has $a_n = 1$ about 41.5% of the time. But our computation makes the asymmetry quantitative at the level of Hausdorff dimension:

• Removing digit 1 from $\{1, \ldots, 15\}$ costs dimension $0.206$
• Removing digit 15 from $\{1, \ldots, 15\}$ costs dimension $0.004$
• The ratio is approximately 50:1

Note: with $N = 15$ Chebyshev collocation nodes, the expected truncation error is on the order of $10^{-3}$. The 0.004 drop for digit 15 is close to this error floor, so the 50:1 ratio should be treated as approximate. A convergence study at higher $N$ (e.g., $N = 25, 35$) for representative alphabets is needed to confirm the precise magnitude of small dimension drops.

This is not merely a curiosity. The dimension of $E_A$ governs the metric theory of Diophantine approximation restricted to digit set $A$: Jarník-type theorems, Khintchine-type dichotomies, and the distribution of rationals with bounded partial quotients all depend on $\dim_H(E_A)$. The extreme dominance of small digits — particularly digit 1 — means that for most applications, the first few digits carry nearly all the information.

Key Results

Five digits with 1 beat fourteen without

Digit set	Cardinality	$\dim_H(E_A)$
$\{1, 2, 3, 4, 5\}$	5	0.837
$\{2, 3, \ldots, 15\}$	14	0.747
$\{1, 2, \ldots, 15\}$	15	0.953

Dimension cost of removing each digit from $\{1, \ldots, 15\}$

Digit removed	$\dim_H(E_{\{1,\ldots,15\} \setminus \{a\}})$	Cost $\Delta$
1	0.747	0.206
2	0.907	0.046
3	0.932	0.021
4	0.941	0.012
5	0.945	0.008
10	0.951	0.002
15	0.949	0.004

Subsets containing 1 dominate at every cardinality

For every cardinality $k$ from 1 to 14, subsets of $\{1, \ldots, 15\}$ that contain digit 1 have substantially higher average Hausdorff dimension than subsets that do not. The highest-dimension subset of any given size is always the set of lowest consecutive digits starting from 1: $\{1, 2, \ldots, k\}$.

Gauss measure predicts dimension ranking

The Hausdorff dimension $\dim_H(E_A)$ shows a strong rank-correlation with the sum

$S(A) = \sum_{a \in A} \frac{1}{a^2}$

over digits $a$ in the subset $A$. This is expected from the first-order term in the pressure expansion around $s = 1$, though published studies (Hensley 1992; Falk-Nussbaum 2019) show noticeable deviations for mixed alphabets. A formal correlation coefficient (Kendall $\tau$ or Spearman $\rho$) across all subsets has not yet been computed. Since $1/1^2 = 1$ while $1/15^2 \approx 0.004$, this explains qualitatively why digit 1 contributes so disproportionately: the Gauss measure weight $1/a^2$ drops by a factor of 225 from $a = 1$ to $a = 15$.

Empirical growth law

The dimension of consecutive-digit sets follows the empirical fit:

$\dim_H(E_{\{1,\ldots,n\}}) \approx 1 - \frac{0.58}{n^{0.88}}$

This is an empirical fit over the range $n = 1$ to $20$; with only 2—3 significant digits per dimension value and a limited range, many functional forms (including the classical asymptotic $1 - c/\log n$) could fit comparably. Residuals and goodness-of-fit statistics have not been computed. The exponent $0.88$ should be treated as a rough guide, not a precise measurement.

Method

• Transfer operator: $\mathcal{L}_s f(x) = \sum_{a \in A} \frac{1}{(a + x)^{2s}} f\!\left(\frac{1}{a + x}\right)$
• Chebyshev collocation at $N = 15$ nodes on $[0, 1]$
• Hausdorff dimension computed as the unique $s > 0$ where $\lambda_1(\mathcal{L}_s) = 1$
• All $2^{15} - 1 = 32{,}767$ non-empty subsets of $\{1, \ldots, 15\}$ enumerated
• Hardware: NVIDIA RTX 5090

Status

CONFIRMED at n=20. The full computation of all $2^{20} - 1 = 1{,}048{,}575$ subsets completed in 4,343 seconds on the RTX 5090. The dominance pattern not only persists but strengthens:

Digit set	$\dim_H$	Note
$E_{\{1,\ldots,5\}}$	0.837	5 digits with 1
$E_{\{2,\ldots,20\}}$	0.768	19 digits without 1
$E_{\{1,\ldots,20\}}$	0.965	All 20 digits

Five digits with 1 beat nineteen digits without 1 by a margin of 0.069 in Hausdorff dimension. Removing digit 1 from $\{1, \ldots, 20\}$ costs dimension $0.197$ while removing digit 20 costs $0.002$ — a ratio of approximately 100:1 (up from 50:1 at $n = 15$). As with the $n = 15$ case, the 0.002 drop is at the edge of the $N = 15$ collocation truncation error, so the exact ratio should be treated as approximate pending a higher-$N$ convergence study.

Correction (2026-04-01): $\dim_H(E_{\{2,\ldots,20\}})$ was previously reported as 0.826. MCP peer review (Claude Opus 4.6, Anthropic) cross-checked against the actual spectrum data (spectrum_n20.csv) and found the correct value is 0.768. This correction strengthens the finding: the gap between 5-with-1 and 19-without-1 is 0.069, larger than the previously reported 0.011. Digit 1 dominance is even more extreme than originally stated.

Connection to Other Findings

• Zaremba spectral gaps: The dimension $\dim_H(E_{\{1,\ldots,5\}}) = 0.837$ matches the Zaremba semigroup dimension — see finding
• Hausdorff dimension spectrum: This finding is extracted from the full $n = 20$ spectrum — see experiment
• Lyapunov exponent spectrum: The twin dataset of Lyapunov exponents for all $2^{20} - 1$ subsets shows the same dominance pattern — see experiment
• Transfer operator machinery: Same Chebyshev collocation code used for both dimension computation and spectral gap analysis — see experiment

Code

• Transfer operator: scripts/experiments/zaremba-transfer-operator/transfer_operator.cu

References

• Hensley, D. (1992). “Continued fraction Cantor sets, Hausdorff dimension, and functional analysis.” Journal of Number Theory, 40(3), pp. 336–358.
• Jenkinson, O. and Pollicott, M. (2001). “Computing the dimension of dynamically defined sets: E_2 and bounded continued fraction entries.” Ergodic Theory and Dynamical Systems, 21(5), pp. 1429–1445.
• Jarník, V. (1929). “Zur metrischen Theorie der diophantischen Approximationen.” Prace Matematyczno-Fizyczne, 36, pp. 91–106.
• Hausdorff, F. (1919). “Dimension und äußeres Maß.” Mathematische Annalen, 79, pp. 157–179.

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Computed on NVIDIA RTX 5090. All eigenvalue problems solved on GPU via cuSOLVER.

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.