Acta Mathematica Hungarica

, Volume 139, Issue 3, pp 208–227 | Cite as

Purely periodic expansions in systems with negative base

  • Zuzana Masáková
  • Edita Pelantová


We study the question of pure periodicity of expansions in the negative base numeration system. In analogy of Akiyama’s result for positive Pisot unit base β, we find a sufficient condition so that there exist an interval J containing the origin such that the (−β)-expansion of every rational number from J is purely periodic. We focus on the case of quadratic bases and demonstrate the following difference between the negative and positive bases: It is known that the finiteness property (\(\mathop {\mathrm {Fin}}\nolimits \, (\beta)= {\mathbb {Z}}[\beta]\)) is not only sufficient, but also necessary in the case of positive quadratic and cubic bases. We show that \(\mathop {\mathrm {Fin}}\nolimits \, (-\beta)= {\mathbb {Z}}[\beta]\) is not necessary in the case of negative bases.

Key words and phrases

negative base periodic expansions Pisot numbers 

Mathematics Subject Classification

11K16 11A63 


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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2012

Authors and Affiliations

  1. 1.Department of Mathematics FNSPECzech Technical University in PraguePraha 2Czech Republic

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