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

Shigeki Akiyama 1 Guy Barat 2 Valerie Berthe 3 Anne Siegel 4
3 ARITH - Arithmétique informatique
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier
4 SYMBIOSE - Biological systems and models, bioinformatics and sequences
IRISA - Institut de Recherche en Informatique et Systèmes Aléatoires, Inria Rennes – Bretagne Atlantique
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 , so-called x-tiles are introduced, so that 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 on the non-Archimedean completions of the number field TeX corresponding to the prime divisors of the norm of β. This representation space has numerous potential implications. We focus here on 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. The key point relies on the description of the boundary of the tiles in terms of paths on a graph called "boundary graph". The paper ends with explicit quadratic examples, showing that the general behaviour of γ(β) is slightly more complicated than in the unit case.
Type de document :
Article dans une revue
Monatshefte für Mathematik, Springer Verlag, 2008, 155 (3-4), pp.377-419. 〈10.1007/s00605-008-0009-7〉
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https://hal.inria.fr/inria-00330567
Contributeur : Anne Siegel <>
Soumis le : mardi 14 octobre 2008 - 18:06:58
Dernière modification le : jeudi 11 janvier 2018 - 06:26:07

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Shigeki Akiyama, Guy Barat, Valerie Berthe, Anne Siegel. Boundary of central tiles associated with Pisot beta-numeration and purely periodic expansions. Monatshefte für Mathematik, Springer Verlag, 2008, 155 (3-4), pp.377-419. 〈10.1007/s00605-008-0009-7〉. 〈inria-00330567〉

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