Uniform distribution mod $1$ for sequences of ergodic sums and continued fractions

TL;DR

This paper characterizes when ergodic sums are uniformly distributed mod 1 and shows that for almost every real number, the denominators of continued fraction convergents follow Benford's law, with applications to Gibbs-Markov maps.

Contribution

It provides a coboundary condition for uniform distribution mod 1 of ergodic sums and applies it to continued fractions and Gibbs-Markov maps.

Findings

Ergodic sums are uniformly distributed mod 1 under a specific coboundary condition.

Denominators of continued fraction convergents follow Benford's law for almost all real numbers.

Application of results to sequences generated by Gibbs-Markov maps.

Abstract

We establish a coboundary condition for a sequence of ergodic sums (i.e.~Birkhoff partial sums) to be almost surely uniformly distributed mod $1$. Applications are given when the sequence is generated by a Gibbs-Markov map. In particular, we show that for almost every real number, the sequence of denominators of the convergents of its continued fraction expansion satisfies Benford's law.