On the number of unique expansions in non-integer bases

Abstract

Let $q>1$ be a real number and let $m=m(q)$ be the largest integer smaller than $q$. It is well known that each number $x\in J_q:=[0,{\sum }_{i=1}^{\infty }m{q}^{-i}]$ can be written as $x={\sum }_{i=1}^{\infty }{c}_i{q}^{-i}$ with integer coefficients $0\le c_i<q$. If $q$ is a non-integer, then almost every $x\in J_q$ has continuum many expansions of this form. In this note we consider some properties of the set ${\mathcal{U}}_q$ consisting of numbers $x\in J_q$ having a unique representation of this form. More specifically, we compare the size of the sets ${\mathcal{U}}_q$ and ${\mathcal{U}}_r$ for values $q$ and $r$ satisfying $1<q<r$ and $m(q)=m(r)$.

typo corrected in Theorem 1.1

References in corpus (1)

• Greedy and quasi-greedy expansions in non-integer bases