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lirmm-00345407, version 1

## 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

Monatshefte fur Mathematik 155 (2008) 377-419

Abstract: This paper studies tilings and representation sapces related to the $\beta$-transformation when $\beta$ 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 $\beta$-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 $\beta$-expansion, called central tile: for elements $x$ of the ring $\GZ[1/\beta]$, 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 ${\mathbb Q} (\beta)$ corresponding to the prime divisors of the norm of $\beta$. This representation space has numerous potential implications. We focus here on the gamma function $\gamma(\beta)$ defined as the supremum of the set of elements $v$ in $[0,1]$ such that every positive rational number $p/q$, with $p/q \leq v$ and $q$ coprime with the norm of $\beta$, has a purely periodic $\beta$-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 papers ends with explicit quadratic examples, showing that the general behaviour of $\gamma(\beta)$ is slightly more complicated than in the unit case.

• 1:  Department of Mathematics
• Niigata University
• 2:  Institut fur Mathematik
• Technische Universitat Graz
• 3:  Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier (LIRMM)
• CNRS : UMR5506 – Université Montpellier II - Sciences et techniques
• 4:  SYMBIOSE (INRIA - IRISA)
• CNRS : UMR6074 – INRIA – Institut National des Sciences Appliquées (INSA) - Rennes – Université de Rennes 1
• Domain : Mathematics/Number Theory

• lirmm-00345407, version 1
• oai:hal-lirmm.ccsd.cnrs.fr:lirmm-00345407
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• Submitted on: Tuesday, 9 December 2008 10:02:29
• Updated on: Thursday, 11 December 2008 16:10:46