Boundary of central tiles associated with Pisot beta-numeration and purely periodic expansions

TL;DR

This paper investigates tilings linked to Pisot beta-transformations and explores the set of rational numbers with purely periodic beta-expansions, emphasizing quadratic cases and their properties.

Contribution

It provides new insights into tilings associated with non-unit Pisot numbers and characterizes purely periodic beta-expansions for these cases.

Findings

Characterization of tilings related to Pisot beta-transformations

Description of the set of rationals with purely periodic beta-expansions

Analysis of quadratic examples and their properties

Abstract

This paper studies tilings related to the beta-transformation when beta is a Pisot number (that is not supposed to be a unit). Then it applies the obtained results to study the set of rational numbers having a purely periodic beta-expansion. Special focus is given to some quadratic examples.