### Fisher, AM and Zhang, X (2023)

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

Preprint arXiv:2307.14843 [math.DS]; last accessed August 5, 2023
.

**ISSN/ISBN:** Not available at this time.
**DOI:** Not available at this time.

**Abstract:** We establish a necessary and sufficient 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.

**Bibtex:**

```
@misc{,
title={Uniform distribution mod $1$ for sequences of ergodic sums and continued fractions},
author={Albert M. Fisher and Xuan Zhang},
year={2023},
eprint={2307.14843},
archivePrefix={arXiv},
primaryClass={math.DS}
}
```

**Reference Type:** Preprint

**Subject Area(s):** Analysis, Probability Theory