# Finite beta-expansions with negative bases

Abstract : The finiteness property is an important arithmetical property of beta-expansions. We exhibit classes of Pisot numbers $\beta$ having the negative finiteness property, that is the set of finite $(-\beta)$-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 $(-\beta)$-integers. We also give conditions excluding the negative finiteness property.
Document type :
Journal articles
Domain :

https://hal.archives-ouvertes.fr/hal-01620765
Contributor : Wolfgang Steiner <>
Submitted on : Friday, October 20, 2017 - 11:19:07 PM
Last modification on : Friday, January 4, 2019 - 5:33:38 PM

### Citation

Zuzana Krčmáriková, Wolfgang Steiner, Tomáš Vávra. Finite beta-expansions with negative bases. Acta Mathematica Hungarica, Springer Verlag, 2017, 152 (2), pp.485 - 504. ⟨10.1007/s10474-017-0711-9⟩. ⟨hal-01620765⟩

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