# On the number of unique expansions in non-integer bases - Mathematics > Number Theory

Abstract: Let $q > 1$ be a real number and let $m=mq$ be the largest integer smallerthan $q$. 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 integercoefficients $0 \le c i < q$. If $q$ is a non-integer, then almost every $x \inJ q$ has continuum many expansions of this form. In this note we consider someproperties of the set $\mathcal{U} q$ consisting of numbers $x \in J q$ havinga unique representation of this form. More specifically, we compare the size ofthe sets $\mathcal{U} q$ and $\mathcal{U} r$ for values $q$ and $r$ satisfying$1< q < r$ and $mq=mr$.

Author: Martijn de Vries

Source: https://arxiv.org/