Quantitative longest-run laws for partial quotients
Number Theory
2026-02-13 v1 Dynamical Systems
Probability
Abstract
Two longest-run statistics are studied: the longest run of a fixed value and the longest run over all values. Under quantitative mixing and exponential cylinder estimates for constant words, a general theorem is proved. Quantitative almost-sure logarithmic growth is obtained, and eventual two-sided bounds with double-logarithmic error terms are established. For continued-fraction partial quotients, explicit centring constants and double-logarithmic error bounds are derived for both statistics.
Keywords
Cite
@article{arxiv.2602.11462,
title = {Quantitative longest-run laws for partial quotients},
author = {Ying Wai Lee},
journal= {arXiv preprint arXiv:2602.11462},
year = {2026}
}