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Pré-Publication, Document De Travail Année : 2023

Dumont--Thomas numeration systems for $\mathbb{Z}$

Résumé

We extend the well-known Dumont--Thomas numeration system to $\mathbb{Z}$ by considering two-sided periodic points of a substitution, thus allowing us to represent any integer in $\mathbb{Z}$ by a finite word (starting with $\mathtt{0}$ when nonnegative and with $\mathtt{1}$ when negative). We show that an automaton returns the letter at position $n\in\mathbb{Z}$ of the periodic point when fed with the representation of $n$. The numeration system naturally extends to $\mathbb{Z}^d$. We give an equivalent characterization of the numeration system in terms of a total order on a regular language. Lastly, using particular periodic points, we recover the well-known two's complement numeration system and the Fibonacci analogue of the two's complement numeration system.

Dates et versions

hal-04245645 , version 1 (17-10-2023)

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Citer

Sébastien Labbé, Jana Lepšová. Dumont--Thomas numeration systems for $\mathbb{Z}$. 2023. ⟨hal-04245645⟩

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