Spectrum, algebraicity and normalization in alternate bases
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The first aim of this article is to give information about the algebraic properties of alternate bases β=(β_0,…,β_p-1) determining sofic systems. We show that a necessary condition is that the product δ=∏_i=0^p-1β_i is an algebraic integer and all of the bases β_0,…,β_p-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 β_0,…,β_p-1∈ℚ(δ), then the system associated with the alternate base β=(β_0,…,β_p-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=(β_0,…,β_p-1) such that δ is a Pisot number and β_0,…,β_p-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.
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