The digit exchanges in the beta expansion of algebraic numbers

Abstract

In this article, we investigate the $\beta$-expansions of real algebraic numbers. In particular, we give new lower bounds for the number of digit exchanges in the case where $\beta$ is a Pisot or Salem number. Moreover, we define a new class of algebraic numbers, quasi-Pisot numbers and quasi-Salem numbers, which gives a generalization of Pisot numbers and Salem numbers. Our method for the number of digit exchanges is also applicable to more general representations of complex algebraic numbers $\xi$ by infinite series $\xi=\sum_{n=1}^{\infty} t_n \beta^{-n}$, where $\bold{t}=(t_n)_{n\ge 1}\in \Z^{\N}$ is a bounded sequence of integers and $\beta$ is a quasi-Pisot or quasi-Salem number.Comment: 12 page