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

TL;DR

The paper develops a coboundary-based criterion for when a sequence of ergodic sums $\mathcal{S}_n f$ is almost surely uniformly distributed mod $1$, using a circle skew product and Furstenberg-type ergodicity arguments. In the Gibbs-Markov setting, it provides verifiable conditions ensuring the coboundary obstruction cannot hold for natural observables, and applies this to observables related to continued fractions. As a key application, it shows that for almost every $x$, the sequence $\{\,\log q_n(x)\}$ is uniformly distributed mod $1$, which implies Benford behavior for the denominators $q_n(x)$ of continued fraction convergents; this extends known results beyond quadratic irrationals. The method yields a broader mechanism for Benford phenomena in dynamical settings and establishes concrete asymptotics for auxiliary quantities appearing in the approximation of $\log q_n(x)$.

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.

Theorems & Definitions (20)

• Theorem 1.1
• Theorem 2.1
• proof
• Proposition 2.2
• proof
• Definition 2.3
• Proposition 2.4
• proof
• Proposition 3.1
• proof