On alpha-adic expansions in Pisot bases

Abstract

We study $α$-adic expansions of numbers in an extension field, that is to say, left infinite representations of numbers in the positional numeration system with the base $α$, where $α$ is an algebraic conjugate of a Pisot number $β$. Based on a result of Bertrand and Schmidt, we prove that a number belongs to $\mathbb{Q}(α)$ if and only if it has an eventually periodic $α$-expansion. Then we consider $α$-adic expansions of elements of the extension ring $\mathbb{Z}[α^{-1}]$ when $β$ satisfies the so-called Finiteness property (F). In the particular case that $β$ is a quadratic Pisot unit, we inspect the unicity and/or multiplicity of $α$-adic expansions of elements of $\mathbb{Z}[α^{-1}]$. We also provide algorithms to generate $α$-adic expansions of rational numbers.