# The digit exchanges in the rotational beta expansions of algebraic numbers

## TL;DR

This paper studies digit exchanges in beta-expansions of algebraic numbers, introducing new bounds and a generalized class of algebraic numbers, with applications to complex algebraic numbers and specific expansion types.

## Contribution

It introduces quasi-Pisot and quasi-Salem numbers, extending the class of algebraic numbers for beta-expansions and provides new lower bounds for digit exchanges.

## Key findings

- New lower bounds for digit exchanges with Pisot or Salem bases
- Generalization of Pisot and Salem numbers to quasi-Pisot and quasi-Salem
- Application to complex algebraic number expansions

## 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 is applicable also to the digit expansions of complex algebraic numbers, which gives new estimation. In particular, we investigate the digits of rotational beta expansion by Akiyama and Caalim $[3]$ and zeta-expansion by Surer $[20]$, where the base is a quasi-Pisot or quasi-Salem number.