On periodic representations in non-Pisot bases

TL;DR

This paper investigates conditions under which algebraic bases allow for all elements of their number field to have eventually periodic expansions with digits in a finite set, extending known results beyond Pisot bases.

Contribution

It proves a general existence theorem for such digit sets in non-Pisot bases and characterizes bases that admit representations with bounded maximal power proportional to the number's size.

Findings

Existence of digit sets for periodic representations in non-Pisot bases.

If all elements have proportional representations, the base must be Pisot or Salem.

Generalizes Schmidt's result on number representations.

Abstract

We study periodic expansions in positional number systems with a base $\beta\in\C,\ |\beta|>1$, and with coefficients in a finite set of digits $\A\subset\C.$ We are interested in determining those algebraic bases for which there exists $\A\subset \Q(\beta),$ such that all elements of $\Q(\beta)$ admit at least one eventually periodic representation with digits in $\A$. In this paper we prove a general result that guarantees the existence of such an $\A$. This result implies the existence of such an $\A$ when $\beta$ is a rational number or an algebraic integer with no conjugates of modulus $1$. We also consider eventually periodic representations of elements of $\Q(\beta)$ for which the maximal power of the representation is proportional to the absolute value of the represented number, up to some universal constant. We prove that if every element of $\Q(\beta)$ admits such a representation then $\beta$ must be a Pisot number or a Salem number. This result generalises a well known result of Schmidt \cite{Schmidt}.