Functiones et Approximatio Commentarii Mathematici

Numbers with integer expansion in the numeration system with negative base

Petr Ambrož, Daniel Dombek, Zuzana Masáková, and Edita Pelantová

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In this paper, we study representations of real numbers in the positional numeration system with negative base, as introduced by Ito and Sadahiro. We focus on the set $\mathbb{Z}_{-\beta}$ of numbers whose representation uses only non-negative powers of $-\beta$, the so-called $(-\beta)$-integers. We describe the distances between consecutive elements of $\mathbb{Z}_{-\beta}$. In case that this set is non-trivial we associate to $\beta$ an infinite word $\boldsymbol{v}_{-\beta}$ over an (in general infinite) alphabet. The self-similarity of $\mathbb{Z}_{-\beta}$, i.e., the property $-\beta \\mathbb{Z}_{-\beta}\subset \mathbb{Z}_{-\beta}$, allows us to find a~morphism under which $\boldsymbol{v}_{-\beta}$ is invariant. On the example of two cubic irrational bases $\beta$ we demonstrate the difference between Rauzy fractals generated by $(-\beta)$-integers and by $\beta$-integers.

Article information

Funct. Approx. Comment. Math., Volume 47, Number 2 (2012), 241-266.

First available in Project Euclid: 20 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11K16: Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. [See also 11A63]
Secondary: 68R15: Combinatorics on words

numeration system negative base Pisot numbers morphism


Ambrož, Petr; Dombek, Daniel; Masáková, Zuzana; Pelantová, Edita. Numbers with integer expansion in the numeration system with negative base. Funct. Approx. Comment. Math. 47 (2012), no. 2, 241--266. doi:10.7169/facm/2012.47.2.8.

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