Poisson genericity in numeration systems with exponentially mixing probabilities
Probability
2025-07-16 v2 Number Theory
Abstract
We define Poisson genericity for infinite sequences in any finite or countable alphabet with an invariant exponentially-mixing probability measure. A sequence is Poisson generic if the number of occurrences of blocks of symbols asymptotically follows a Poisson law as the block length increases. We prove that almost all sequences are Poisson generic. Our result generalizes Peres and Weiss' theorem about Poisson genericity of integral bases numeration systems. In particular, we obtain that their continued fraction expansions for almost all real numbers are Poisson generic.
Cite
@article{arxiv.2411.04116,
title = {Poisson genericity in numeration systems with exponentially mixing probabilities},
author = {Nicolás Álvarez and Verónica Becher and Eda Cesaratto and Martín Mereb and Yuval Peres and Benjamin Weiss},
journal= {arXiv preprint arXiv:2411.04116},
year = {2025}
}