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Boundary of central tiles associated with Pisot beta-numeration and purely periodic expansions. (English) Zbl 1190.11005
This paper under review studies tilings and representation spaces related to the $$\beta$$-transformation when $$\beta$$ is a Pisot number, that is not a unit. The obtained results are applied to study the set of rational numbers having a purely periodic $$\beta$$-expansion. The authors make use of the connection between pure periodicity and a compact self-similar representation of number having no fractional part in their $$\beta$$-expansion, called center tile. For elements $$x$$ of the ring $$\mathbb Z[1/\beta]$$, 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 the authors working in. This space, called complete representation space, is based on archimedean as well as on the non-archimedean completions of the number field $$\mathbb Q(b)$$ corresponding to the prime divisors of the norm of $$\beta$$. This representation space has numerous potential implications. The authors focus 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 paper ends with explicit quadratic examples, showing that the general behaviour of $$\gamma(\beta)$$ is slightly more complicated than in the unit case.

##### MSC:
 11A63 Radix representation; digital problems 03D45 Theory of numerations, effectively presented structures 11S99 Algebraic number theory: local fields 28A75 Length, area, volume, other geometric measure theory 52C23 Quasicrystals and aperiodic tilings in discrete geometry
##### Keywords:
beta-numeration; tilings; periodic expansions
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