Integers in number systems with positive and negative quadratic Pisot base

Department of Mathematics FNSPE, Czech Technical University in Prague, Trojanova 13, 12000 Praha 2, Czech Republic.
zuzana.masakova@fjfi.cvut.cz; t.vavra@seznam.cz

Received: 14 March 2014
Accepted: 17 March 2014

Abstract

We consider numeration systems with base β and − β, for quadratic Pisot numbers β and focus on comparing the combinatorial structure of the sets Z_β and Z_{− β} of numbers with integer expansion in base β, resp. − β. Our main result is the comparison of languages of infinite words u_β and u_{− β} coding the ordering of distances between consecutive β- and (− β)-integers. It turns out that for a class of roots β of x² − mx − m, the languages coincide, while for other quadratic Pisot numbers the language of u_β can be identified only with the language of a morphic image of u_{− β}. We also study the group structure of (− β)-integers.