On periodic representations in non-Pisot bases

Abstract

We study periodic expansions in positional number systems with a base $β\in\C,\ |β|>1$, and with coefficients in a finite set of digits $\A\subset\C.$ We are interested in determining those algebraic bases for which there exists $\A\subset \Q(β),$ such that all elements of $\Q(β)$ 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 $β$ is a rational number or an algebraic integer with no conjugates of modulus $1$. We also consider eventually periodic representations of elements of $\Q(β)$ 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 $\Q(β)$ admits such a representation then $β$ must be a Pisot number or a Salem number. This result generalises a well known result of Schmidt \cite{Schmidt}.