Periodic representations in algebraic bases

TL;DR

This paper investigates periodic number representations in algebraic bases, establishing conditions under which all elements of certain number fields have eventually periodic expansions with a finite digit set.

Contribution

It proves that if an algebraic base has no Galois conjugate on the unit circle, then all elements of its number field have eventually periodic representations with a finite digit alphabet.

Findings

Existence of finite digit alphabet for periodic representations

Condition on algebraic base's Galois conjugates

Applicability to elements of the number field

Abstract

We study periodic representations in number systems with an algebraic base $\beta$ (not a rational integer). We show that if $\beta$ has no Galois conjugate on the unit circle, then there exists a finite integer alphabet $\mathcal A$ such that every element of $\mathbb Q(\beta)$ admits an eventually periodic representation with base $\beta$ and digits in $\mathcal A$.