Equidistribution of digits in powers and Diophantine approximations

TL;DR

This paper explores how the distribution of digits in powers of integers relates to the Diophantine properties of their logarithms, shedding light on conjectures about digit equidistribution.

Contribution

It establishes a connection between digit equidistribution in base-$b$ expansions of $a^n$ and the Diophantine nature of $ heta_{a,b}=rac{ ext{log}_b a$}, offering new insights into their interplay.

Findings

Digit distribution conjecture linked to Diophantine approximation properties.

Established conditions under which digit equidistribution occurs.

Identified implications for the nature of $ heta_{a,b}$ based on digit patterns.

Abstract

Given integers $a,b>1$ with $\log_b{a}$ irrational, we investigate the connection between the conjectured asymptotic equidistribution of digits in the base-$b$ expansion of $a^n$ and the (non-)Diophantine properties of the number $\theta_{a,b}=\log_b{a}$.