On the Lebesgue measure of sum-level sets for continued fractions

TL;DR

This paper analyzes the measure-theoretic properties of sum-level sets in continued fractions, confirming their measure decays to zero and providing precise asymptotic estimates, linking ergodic theory and number theory.

Contribution

It proves a conjecture that the Lebesgue measure of sum-level sets tends to zero and derives detailed asymptotic decay rates using recent ergodic theory advances.

Findings

Lebesgue measure of sum-level sets decays to zero as level increases

Provided precise asymptotic estimates for the decay rate

Connected ergodic theory results with number theory applications

Abstract

In this paper we give a detailed measure theoretical analysis of what we call sum-level sets for regular continued fraction expansions. The first main result is to settle a recent conjecture of Fiala and Kleban, which asserts that the Lebesgue measure of these level sets decays to zero, for the level tending to infinity. The second and third main result then give precise asymptotic estimates for this decay. The proofs of these results are based on recent progress in infinite ergodic theory, and in particular, they give non-trivial applications of this theory to number theory. The paper closes with a discussion of the thermodynamical significance of the obtained results, and with some applications of these to metrical Diophantine analysis.