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arXiv · 2602.11462

Quantitative longest-run laws for partial quotients

Ying Wai Lee
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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.

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Ying Wai Lee. 2026-02-12. Quantitative longest-run laws for partial quotients. https://arxiv.org/abs/2602.11462

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