Quantitative longest-run laws for partial quotients
Ying Wai Lee
TL;DR
This work studies the longest-run statistics for partial quotients of continued fractions under Gauss measure. It develops a general theorem: for shift-invariant processes with quantitative mixing and exponential cylinder estimates, the longest-run lengths $L_n$ and $R_n$ have almost-sure growth on the scale $\log n /(2\log \rho)$ with a double-logarithmic additive error. It then verifies the hypotheses in the Gauss system, deriving explicit constants $\tau(\lambda)$ and $\varphi$ that govern fixed-symbol and maximised runs: $|L_n(x,\lambda)-\frac{\log n}{2\log \tau(\lambda)}|\le \frac{c\log\log n}{\log \tau(\lambda)}$ and $|R_n(x)-\frac{\log n}{2\log \varphi}|\le \frac{c\log\log n}{\log \varphi}$. Overall, the paper sharpens earlier first-order laws by providing effective rate refinements and offers a general framework for quantitative longest-run analysis in dynamical-number-theoretic settings.
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.