Uniform distribution mod $1$ for sequences of ergodic sums and continued fractions
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866909515472961536 |
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| author | Fisher, Albert M. Zhang, Xuan |
| author_facet | Fisher, Albert M. Zhang, Xuan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_14843 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Uniform distribution mod $1$ for sequences of ergodic sums and continued fractions Fisher, Albert M. Zhang, Xuan Dynamical Systems 11K50, 11K06, 37A50 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. |
| title | Uniform distribution mod $1$ for sequences of ergodic sums and continued fractions |
| topic | Dynamical Systems 11K50, 11K06, 37A50 |
| url | https://arxiv.org/abs/2307.14843 |