Extremal properties of (epi)Sturmian sequences and distribution modulo $1$

Abstract

Starting from a study of Y. Bugeaud and A. Dubickas (2005) on a question about the distribution of real numbers modulo $1$ {\em via\,} combinatorics on words, we survey some combinatorial properties of (epi)Sturmian sequences and distribution modulo $1$ in connection to their work. In particular we focus on extremal properties of (epi)Sturmian sequences, some of which have been rediscovered several times.