Validated numerics for the Gauss problem on continued fractions

Abstract: We get estimates on the Gauss-Kuzmin-Wirsing constant for continued fractions.

• The paper achieves the strongest rigorous bounds on the Gauss-Kuzmin-Wirsing constant using a validated min-max principle and Chebyshev polynomial interpolation.
• It employs a transfer operator framework, conjugating to an operator amenable to Hurwitz zeta functions, to certify the numerical estimates with rigorous error control.
• The methodology also extends to multifractal problems by precisely estimating the Hausdorff dimension of Cantor sets of continued fractions.

Validated Numerics for the Gauss Problem on Continued Fractions

Introduction and Historical Context

The Gauss problem on continued fractions addresses the statistical properties of the tails in the continued fraction expansions of irrationals in $(0,1)$. The central issue, originating from Gauss's own notebooks and correspondence, is to characterize the limiting distribution of the sequence $(\xi_n)$, where $\xi_n$ denotes the $n$th tail of the continued fraction. Gauss claimed that the asymptotic distribution function $F_n(x)$ satisfies

$$\lim_{n \to \infty} F_n(x) = \frac{\log(1+x)}{\log 2},$$

but did not provide a satisfactory proof. Rigorous solutions, spearheaded by works of Kuzmin and Lévy, established exponential convergence to this limit, with the rate governed by the absolute value of the second eigenvalue $\lambda_2$ of a certain transfer operator associated with the Gauss map. Over the years, refined estimates of $\lambda_2$ (the so-called Gauss-Kuzmin-Wirsing constant) have been obtained, culminating in high-precision but, in some cases, non-rigorous numerical results. This paper addresses the need for validated and easily checkable rigorous estimates for $\lambda_2$.

The Transfer Operator Framework

The statistical behavior of the Gauss map $T(x) = \{1/x\}$ on $(\xi_n)$0 and associated distributions is encoded via its transfer operator $(\xi_n)$1 acting on a space of analytic functions. Specifically, on a Banach space $(\xi_n)$2 of analytic functions defined on a suitable disk and subject to the normalization condition with respect to the invariant Gauss measure, $(\xi_n)$3 is defined as

$(\xi_n)$4

This operator has a simple maximal eigenvalue at $(\xi_n)$5, corresponding to the Gauss measure, and a simple second eigenvalue $(\xi_n)$6 responsible for the exponential rate of convergence in the Gauss-Kuzmin theorem.

The analysis is conducted on the cone of functions with non-negative derivatives, enabling the application of Krasnoselskii theory for positive operators in a Banach space, which guarantees the simplicity and spectral isolation of the relevant eigenvalues.

Min-Max Principle and Rigorous Bounds

The operator $(\xi_n)$7 is conjugated to $(\xi_n)$8, which preserves constants and is more amenable to analysis on normalized function spaces. The action of $(\xi_n)$9 on polynomials and monomials is described using Hurwitz zeta functions $\xi_n$0, providing a tractable means for spectral computations.

A pivotal aspect of the approach is a min-max principle for $\xi_n$1. For any function $\xi_n$2 in the positive cone,

$\xi_n$3

This result, adapted from Mayer-Roepstorff, underpins the construction of easily validated, rigorous numerical bounds.

Lagrange-Chebyshev Polynomial Interpolation and Hurwitz Zeta Functions

To obtain practical bounds with controlled and validated errors, the method uses Lagrange interpolation at Chebyshev points. The transfer operator $\xi_n$4 is represented as a finite-rank matrix using basis polynomials. The eigenproblem for this matrix yields a polynomial eigenvector $\xi_n$5 corresponding to the sought extremal eigenvalue. The analyticity of all expressions and the explicit representation via Hurwitz zeta functions ensure rigorous floating-point error control.

Strong numerical results are achieved for $\xi_n$6, with the bounds tightening dramatically with increasing polynomial degree. For example, with $\xi_n$7 (corresponding to degree-79 polynomials), the resulting rigorous bounds on $\xi_n$8 agree to $\xi_n$9 decimal places: $n$0 The approach easily extends to even higher degrees as computational resources allow, with all steps susceptible to validation using interval arithmetic.

Further Applications: Hausdorff Dimension of Cantor Sets of Continued Fractions

The latter sections of the paper consider the Hausdorff dimension $n$1 of sets of numbers with continued fraction expansions constrained to entries in a finite set $n$2. Using similar transfer operator methods and rigorous min-max bounds in polynomial function spaces, dimension estimates are obtained with precision up to $n$3 decimal places for various parameter sets $n$4. This demonstrates the flexibility and power of the method for other multifractal problems within continued fractions and dynamical systems.

Implications and Future Directions

This work makes the strongest to-date validated, high-precision estimates of the Gauss-Kuzmin-Wirsing constant $n$5, providing a standard for future computational and theoretical investigations into limit laws for continued fractions, ergodic properties of the Gauss map, and spectral theory of transfer operators. The approach bridges the gap between fast, but non-rigorous, linear algebraic approximations (e.g., matrix truncations) and the need for mathematically certified results.

The use of analyticity, interpolation at Chebyshev points, and effective utilization of special function representations (Hurwitz zeta) in operator computation paves the way for further applications. This includes eigenvalue estimation in a wide class of dynamical systems with transfer operator descriptions, validated numerical calculation for multifractal spectra, and potential generalizations to higher rank or non-uniformly hyperbolic cases.

Conclusion

The paper develops a validated, efficient, and fully rigorous framework for numerically analyzing the Gauss problem on continued fractions by leveraging transfer operator theory, min-max spectral bounds, analytic polynomial interpolation, and Hurwitz zeta functions. The results set a new standard for precision and mathematical rigor in this classical area and provide tools applicable to broader classes of spectral and multifractal problems in ergodic theory and dynamical systems (2606.13958).