Universal $β$-expansions

Abstract

Given $β\in(1,2)$, a $β$-expansion of a real $x$ is a power series in base $β$ with coefficients 0 and 1 whose sum equals $x$. The aim of this note is to study certain problems related to the universality and combinatorics of $β$-expansions. Our main result is that for any $β\in(1,2)$ and a.e. $x\in (0,1)$ there always exists a universal $β$-expansion of $x$ in the sense of Erdös and Komornik, i.e., a $β$-expansion whose complexity function is $2^n$. We also study some questions related to the points having less than a full branching continuum of $β$-expansions and also normal $β$-expansions.