The digit exchanges in the rotational beta expansions of algebraic numbers

Abstract

In this article, we investigate the β-expansions of real algebraic numbers. In particular, we give new lower bounds for the number of digit exchanges in the case where β 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 is applicable also to the digit expansions of complex algebraic numbers, which gives a new estimate. In particular, we investigate the digits of rotational beta expansion considered by Akiyama and Caalim [3] and zeta-expansion by Surer [20], where the base is a quasi-Pisot or quasi-Salem number.