Digit frequencies of beta-expansions

Abstract

Let $\beta >1$ be a non-integer. First we show that Lebesgue almost every number has a $\beta$-expansion of a given frequency if and only if Lebesgue almost every number has infinitely many $\beta$-expansions of the same given frequency. Then we deduce that Lebesgue almost every number has infinitely many balanced $\beta$-expansions, where an infinite sequence on the finite alphabet $\{0,1 , \ldots ,m\}$ is called balanced if the frequency of the digit $k$ is equal to the frequency of the digit $m-k$ for all $k\in \{0,1 , \ldots ,m\}$. Finally we consider variable frequency and prove that for every pseudo-golden ratio $\beta \in (1,2)$, there exists a constant $c=c(\beta )>0$ such that for any $p\in [\frac{1}{2}-c,\frac{1}{2}+c]$, Lebesgue almost every $x$ has infinitely many $\beta$-expansions with frequency of zeros equal to $p$.