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Spectrum, algebraicity and normalization in alternate bases

The first aim of this article is to give information about the algebraic properties of alternate bases β=(β0,…,βp−1) determining sofic systems. We show that a necessary condition is that the product δ=∏i=0p−1βi is an algebraic integer and all of the bases β0,…,βp−1 belong to the algebraic field Q(δ)...

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Bibliographic Details
Published in:Journal of number theory 2023-08, Vol.249, p.470-499
Main Authors: Charlier, Émilie, Cisternino, Célia, Masáková, Zuzana, Pelantová, Edita
Format: Article
Language:English
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Summary:The first aim of this article is to give information about the algebraic properties of alternate bases β=(β0,…,βp−1) determining sofic systems. We show that a necessary condition is that the product δ=∏i=0p−1βi is an algebraic integer and all of the bases β0,…,βp−1 belong to the algebraic field Q(δ). On the other hand, we also give a sufficient condition: if δ is a Pisot number and β0,…,βp−1∈Q(δ), then the system associated with the alternate base β=(β0,…,βp−1) is sofic. The second aim of this paper is to provide an analogy of Frougny's result concerning normalization of real bases representations. We show that given an alternate base β=(β0,…,βp−1) such that δ is a Pisot number and β0,…,βp−1∈Q(δ), the normalization function is computable by a finite Büchi automaton, and furthermore, we effectively construct such an automaton. An important tool in our study is the spectrum of numeration systems associated with alternate bases. The spectrum of a real number δ>1 and an alphabet A⊂Z was introduced by Erdős et al. For our purposes, we use a generalized concept with δ∈C and A⊂C and study its topological properties.
ISSN:0022-314X
1096-1658
DOI:10.1016/j.jnt.2023.02.012