 
Summary: BOUNDARY OF CENTRAL TILES ASSOCIATED WITH PISOT
BETANUMERATION 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 selfsimilar representation of numbers
having no fractional part in their expansion, called central tile: for elements x of the ring Z[1/],
socalled xtiles are introduced, so that the central tile is a finite union of xtiles up to translation.
These xtiles 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
nonArchimedean 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.
