Digit frequencies of beta-expansions

Abstract

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