Quantitative longest-run laws for partial quotients

TL;DR

This paper investigates the statistical behavior of the longest runs in sequences of partial quotients, providing precise growth estimates and bounds for continued fractions under certain mixing conditions.

Contribution

It introduces a general theorem for longest-run statistics and derives explicit bounds and constants for continued-fraction partial quotients.

Findings

Almost-sure logarithmic growth of longest runs

Two-sided bounds with double-logarithmic error terms

Explicit constants for continued fractions

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.