Finite beta-expansions with negative bases


DOI: 10.1007/s10474-017-0711-9

Cite this article as:
Krčmáriková, Z., Steiner, W. & Vávra, T. Acta Math. Hungar. (2017). doi:10.1007/s10474-017-0711-9


The finiteness property is an important arithmetical property of beta-expansions. We exhibit classes of Pisot numbers β having the negative finiteness property, that is the set of finite (−β)-expansions is equal to \({\mathbb{Z}[\beta^{-1}]}\). For a class of numbers including the Tribonacci number, we compute the maximal length of the fractional parts arising in the addition and subtraction of (−β)-integers. We also give conditions excluding the negative finiteness property.

Mathematics Subject Classification

11A63 11K16 

Key words and phrases

beta-expansion finiteness shift radix system 

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2017

Authors and Affiliations

  1. 1.Department of Mathematics FNSPECzech Technical University in PraguePraha 2Czech Republic
  2. 2.IRIF, CNRS UMR 8243Université Paris Diderot – Paris 7Paris Cedex 13France

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