Precise asymptotics on the Birkhoff sums for dynamical systems

TL;DR

This paper derives precise asymptotic formulas for Birkhoff sums in dynamical systems, extending classical results and applying them to the Gauss map and continued fractions, revealing new insights into their asymptotic behavior.

Contribution

It provides the first precise asymptotic results for Birkhoff sums in dynamical systems, paralleling classical results for i.i.d. sums, and applies these to the Gauss map and continued fractions.

Findings

New precise asymptotics for Birkhoff sums in dynamical systems

Application to the Gauss map and continued fraction expansions

Extension of classical probabilistic sum results to dynamical contexts

Abstract

We establish two precise asymptotic results on the Birkhoff sums for dynamical systems. These results are parallel to that on the arithmetic sums of independent and identically distributed random variables previously obtained by Hsu and Robbins, Erd\H{o}s, Heyde. We apply our results to the Gauss map and obtain new precise asymptotics in the theorem of L\'evy on the regular continued fraction expansion of irrational numbers in $(0,1)$.