Summary: BOUNDARY OF CENTRAL TILES ASSOCIATED WITH PISOT
BETA-NUMERATION AND PURELY PERIODIC EXPANSIONS
SHIGEKI AKIYAMA, GUY BARAT, VAL´ERIE BERTH´E, AND ANNE SIEGEL
Robert Tichy, aus Anlass seines f¨unfzigsten Geburtstages gewidmet
Abstract. This paper studies tilings and representation sapces related to the -transformation
when 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 -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 -expansion, called central tile: for elements x of the ring Z[1/],
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 Q() corresponding to the prime divisors of the
norm of . This representation space has numerous potential implications.
We focus here on the gamma function () 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 , has a purely periodic -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 () is slightly more complicated than
in the unit case.