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

Abstract. This paper studies tilings and representation sapces related to the β-transformation when β is a Pisot number (that is not supposed to be a unit). The obtained results are applied to study the set of rational numbers having a purely periodic β-expansion. We indeed make use of the connection between pure periodicity and a compact self-similar representation of numbers having no fractional part in their β-expansion, called central tile: for elements x of the ring Z[1/β], so-called x-tiles are introduced, so the central tile is a finite union of x-tiles up to translation. These x-tiles provide a covering (and even in some cases a tiling) of the space we are working in. This space, called complete representation space, is based on Archimedean as well as non-Archimedean completions of the number field Q(β) for primes dividing the norm of β. This representation space has numerous potential implications. We focus here on one application concerning the gamma function γ(β) defined as the supremum of the set of elements v in [0, 1] such that every positive rational number p/q, with p/q ≤ v and q coprime with the norm of β, has a purely periodic β-expansion. Our study relies on the description of the topological properties of central tiles in terms of boundary graphs. Special focus is given to some quadratic examples, showing that the behaviour of γ(β) in the non-unit case is slightly different from its behaviour in the unit case. 1