Digit frequencies of beta-expansions

TL;DR

This paper investigates the frequency properties of beta-expansions for non-integer bases, showing almost every number has infinitely many expansions with prescribed digit frequencies, including balanced and variable frequencies, especially for pseudo-golden ratios.

Contribution

It establishes the equivalence of having a single frequency and infinitely many expansions with that frequency, and proves the existence of infinitely many expansions with balanced and variable digit frequencies for certain bases.

Findings

Almost every number has infinitely many beta-expansions with a given digit frequency.

Existence of infinitely many balanced beta-expansions for Lebesgue almost every number.

For pseudo-golden ratios, almost every number has infinitely many expansions with zero frequency close to 1/2.

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 $\{0,1,\cdots,m\}$ 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 $\beta\in(1,2)$, 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 $p$.