Acta Mathematica Hungarica

, Volume 140, Issue 4, pp 329–340 | Cite as

Optimal number representations in negative base

  • Zuzana Masáková
  • Edita Pelantová


For a given base γ and a digit set \(\mathcal{B}\) we consider optimal representations of a number x, as defined by Dajani et al. [3]. For a non-integer negative base γ=−β<−1 and the digit set \(\mathcal{A}_{\beta}:= \{0,1,\dots,\lceil\beta\rceil-1\}\) we derive the transformation which generates the optimal representation, if it exists. We show that – unlike the case of negative integer base – almost no x has an optimal representation. For a positive base γ=β>1 and the alphabet \(\mathcal{A}_{\beta}\) we provide an alternative proof of statements obtained by Dajani et al.

Key words and phrases

negative base greedy expansion Pisot number 

Mathematics Subject Classification

11K16 11A63 


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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2013

Authors and Affiliations

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

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