On the expansions of real numbers in two multiplicative dependent bases

TL;DR

This paper investigates the combined complexity of the base-$r$ and base-$s$ expansions of irrational numbers when $r$ and $s$ are multiplicatively dependent, establishing an optimal lower bound for their sum of block complexities.

Contribution

It provides the first explicit lower bound for the sum of block complexities of expansions in two multiplicatively dependent bases, demonstrating the bound's optimality.

Findings

Established a lower bound for the sum of block complexities

Proved the bound is best possible

Applied to expansions in multiplicatively dependent bases

Abstract

Let $r \ge 2$ and $s \ge 2$ be multiplicatively dependent integers. We establish a lower bound for the sum of the block complexities of the $r$-ary expansion and of the $s$-ary expansion of an irrational real number, viewed as infinite words on $\{0, 1, \ldots , r-1\}$ and $\{0, 1, \ldots , s-1\}$, and we show that this bound is best possible.