Distributional asymptotics mod 1 of $(\log_bn)$

TL;DR

This paper investigates the distributional asymptotics of the logarithm sequence $( ext{log}_b n)$ modulo one, providing improved convergence rate estimates and establishing sharp bounds for the distribution of leading digits.

Contribution

The paper offers new precise convergence rates for the distribution of $( ext{log}_b n)$ modulo one, including sharp bounds, advancing understanding of its distributional behavior.

Findings

Improved upper estimate $rac{ oot{2}\log N}{N}$ for convergence rate on the circle.

Sharp convergence rate $rac{ ext{log} N}{N}$ on the interval $[0,1]$.

Verification that the discrepancy metric rate $rac{ ext{log} N}{N}$ is optimal.

Abstract

This paper studies the distributional asymptotics of the slowly changing sequence of logarithms $(\log_bn)$ with $b\in\mathbb{N}\setminus\{1\}.$ It is known that $(\log_bn)$ is not uniformly distributed modulo one, and its omega limit set is composed of a family of translated exponential distributions with constant $\log b.$ An improved upper estimate $\left(\sqrt{\log N}/N\right)$ is obtained for the rate of convergence with respect to (w.r.t.) the Kantorovich metric on the circle, compared to the general results on rates of convergence for a class of slowly changing sequences in the author's companion in-progress work. Moreover, a sharp rate of convergence $\left(\log N/N\right)$ w.r.t. the Kantorovich metric on the interval $[0,1]$, is derived. As a byproduct, the rate of convergence w.r.t. the discrepancy metric (or the Kolmogorov metric) turns out to be $\left(\log N/N\right)$ as well, which verifies that an upper bound for this rate derived in [Y. Ohkubo and O. Strauch, Distribution of leading digits of numbers, Unif. Distrib. Theory, $\textbf{11}$ (2016), no.1, 23--45.] is sharp.