Asymptotic statistics for finite continued fractions with restricted digits (2512.11357v1)

Abstract: Zaremba's conjecture concerns a formation of continued fraction expansions for rational numbers with partial quotient bounded by an absolute constant. We present asymptotic estimates for the size of $ε$-thickening of certain fractal sets of bounded-type, which in turn provide additional support for Zaremba's conjecture on average. We also conclude a generalisation for complex continued fractions over imaginary quadratic fields.

• The paper establishes new asymptotic growth rates for ε-thickened rational sets, connecting spectral data with statistics of bounded-digit continued fractions.
• It employs transfer operator techniques and spectral analysis to derive explicit cardinality estimates and improve error terms in both real and complex settings.
• The study provides unconditional bounds supporting Zaremba’s conjecture while outlining the analytical challenges in achieving conjectural exponents for fixed denominators.

Asymptotic Statistics for Finite Continued Fractions with Restricted Digits

Introduction and Context

The paper "Asymptotic statistics for finite continued fractions with restricted digits" (2512.11357) tackles refined quantitative aspects of Zaremba's conjecture using ergodic theory, transfer operator techniques, and spectral analysis. Zaremba's conjecture asserts that every positive integer $N$ admits a reduced fraction $a/N$ whose continued fraction expansion's digits are uniformly bounded by an absolute constant $A$. Despite significant advances (e.g., the Bourgain–Kontorovich theorem for almost all $N$ with $A=50$), the conjecture remains unresolved in full generality.

The author reformulates Zaremba’s conjecture in terms of fractal geometry: in particular, the Hausdorff dimension $\delta_A$ of the set $E_A$ of real numbers in $[0,1]$ with all continued fraction digits $\leq A$. Hensley’s conjecture links the threshold $\delta_A>1/2$ to the distribution of admissible denominators supporting Zaremba’s conjecture infinitely often.

Main Results

The paper's principal achievements are the establishment of precise asymptotic growth rates for "ε-thickened" rational sets defined by bounded partial quotients, both in the classical (real) and complex (imaginary quadratic field) settings. Specifically, the author constructs an auxiliary probability space $\Sigma_{N,A}(\epsilon)$ such that

$$\Sigma_{N,A} \subset \Sigma_{N,A}(\epsilon) \subset \Omega_{N,A}$$

and proves the cardinality estimate

$|\Sigma_{N,A}(\epsilon)| \sim N^{2\delta_A - \eta}$

for some $\eta\in (0,1)$, thus providing non-trivial upper bounds interpolating between the trivial and conjectural exponents and thus adding further averaging evidence for Zaremba's conjecture.

For complex continued fractions over norm-Euclidean imaginary quadratic fields, analogous constructions yield sets $\Sigma_{N, \mathcal{A}_d}(\epsilon)$ with estimates of the form $N^{2\delta_{\mathcal{A}_d} - \theta}$, with explicit discussion of the arithmetic and combinatorial complications and a discussion of the limitations and possible improvements for the complex case.

Methodology and Technical Approach

The technical apparatus is rooted in a dynamical reformulation of the combinatorial counting problems using the transfer operators associated to the Gauss map and its complex analogs. The arguments proceed by:

1. Transfer Operators with Digit Restrictions: For a given maximal digit $A$, a finite family of restricted inverse branches of the Gauss map induces a compact transfer operator $\mathcal{L}_{s, w, A}$ acting on $C^1$-spaces.
2. Spectral Analysis: The operators are shown (via quasi-compactness and analytic perturbation theory) to have a unique simple maximal eigenvalue, with further Dolgopyat-type polynomial decay estimates for the resolvent, essential for Tauberian arguments. In the complex setting, additional Markov partitions and domain decompositions are employed to handle admissibility conditions in the non-commutative lattice structure.
3. Dirichlet Series Generation: Rational counting is encoded via two-variable Dirichlet series $L(s,w)$, connected to the dynamical system via transfer operator traces and resolvents. Analytic continuation and pole analysis of $L(s,w)$ provide asymptotic coefficients through contour integration.
4. Perron's Formula and Smoothing: An adaptation of Perron's formula with explicit error terms allows the extraction of asymptotics for partial sums $\sum_{n \leq N} d_n(w)$. The introduction of an $\epsilon$-thickened interval and a corresponding smoothing process improves error terms and interpolates exponents, ultimately yielding the stated asymptotic formulae for $\epsilon$-enlarged sets (rather than just for fixed $N$).

Numerical and Quantitative Findings

The core quantitative estimates are:

• For the "thickened" set of reduced rationals with denominators in $[N - N^\beta, N]$ (where $\beta \in (0,1)$), the cardinality grows as $N^{2\delta_A - \eta}$ for some $\eta \in (0,1)$ explicitly computable from spectral data and the smoothing parameter.
• In the classical setting, these exponents bridge the "trivial" exponent $2\delta_A$ (for unconstrained averages) and the conjectured minimal exponent $2\delta_A-1$ (Zaremba/Hensley's prediction for pointwise behavior).
• For the complex (imaginary quadratic field) case, the corresponding exponent is $2\delta_{\mathcal{A}_d} - \theta$ with $\theta \in (1,2)$, reflecting the different fractal geometry of these spaces.

Implications and Theoretical Significance

The results provide new unconditional bounds on the statistical distribution of bounded-type rationals and their analogs in imaginary quadratic fields, reinforcing the predictions of Zaremba's conjecture in an averaged or "thickened" sense. The spectral approach strengthens the connection between transfer operator eigenvalue data and Diophantine counting, demonstrating further utility of ergodic and thermodynamic tools in algebraic and analytic number theory. The extension to complex continued fractions underscores the deeper arithmetic nature of these problems and frames new avenues in the paper of Diophantine approximation in higher dimensions and non-commutative settings.

Importantly, the techniques clarify why the exact fixed-denominator case remains challenging: the spectral gaps and smoothing are essential to recover the sharpest exponents, and the failure to fully bridge the gap to $2\delta_A-1$ highlights both analytic and combinatorial obstructions. The methods can potentially influence the paper of exceptional sets in Diophantine problems, Hausdorff dimension calculations for fractal sets, and transfer operator applications across arithmetic dynamics.

Directions for Future Work

The framework opens several directions:

• Improved Exponents: Further refinement of spectral gap and decay estimates (potentially via higher-order Dolgopyat arguments or improved Markov partitions) may yield sharper exponents, possibly reaching the conjectural value.
• Non-Euclidean and Higher Rank Extensions: The complex analysis suggests avenues for generalization to other number fields, possibly via the paper of continued fractions in quaternion algebras or more general semigroups.
• Explicit Error Terms and Distributional Laws: The machinery developed here could adapt to the paper of local statistics (e.g., distribution of partial quotient vectors) or to the finer distributional results such as local limit theorems or large deviation bounds.
• Connection to Thin Groups: The arithmetic and spectral content, particularly as developed by Bourgain–Kontorovich, hints at deeper ties to thin orbits, expander graphs, and spectral gaps for non-lattice actions, which may unlock stronger results on Zaremba-type conjectures.

Conclusion

The paper provides a rigorous and substantial quantitative analysis of the asymptotics of bounded-digit continued fractions. By combining operator theory, analytic number theory, and ergodic dynamical tools, it achieves new intermediate results supporting Zaremba's conjecture and extends these ideas to the context of complex continued fractions. These results provide both new evidence in the "averaged" setting and a flexible analytic framework for further investigation in Diophantine fractal geometry and arithmetic dynamics (2512.11357).