On the expansions of real numbers in two integer bases

TL;DR

This paper proves that for two multiplicatively independent bases, an irrational number cannot have both its base-$r$ and base-$s$ expansions exhibit low complexity, such as both being Sturmian words, highlighting limitations in simultaneous simple expansions.

Contribution

It establishes a new limitation on the complexity of simultaneous expansions of irrationals in multiplicatively independent bases.

Findings

Cannot have both expansions as Sturmian words

Low block complexity cannot occur simultaneously in both expansions

Provides a new link between number theory and combinatorics on words

Abstract

Let $r$ and $s$ be multiplicatively independent positive integers. We establish that the $r$-ary expansion and the $s$-ary expansion of an irrational real number, viewed as infinite words on $\{0, 1, \ldots , r-1\}$ and $\{0, 1, \ldots , s-1\}$, respectively, cannot have simultaneously a low block complexity. In particular, they cannot be both Sturmian words.