Spectrum, algebraicity and normalization in alternate bases

The first aim of this article is to give information about the algebraic properties of alternate bases $\boldsymbolβ=(β_0,\dots,β_{p-1})$ determining sofic systems. We show that a necessary condition is that the product $δ=\prod_{i=0}^{p-1}β_i$ is an algebraic integer and all of the bases $β_0,\ldots,β_{p-1}$ belong to the algebraic field ${\mathbb Q}(δ)$. On the other hand, we also give a sufficient condition: if $δ$ is a Pisot number and $β_0,\ldots,β_{p-1}\in {\mathbb Q}(δ)$, then the system associated with the alternate base $\boldsymbolβ=(β_0,\dots,β_{p-1})$ 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 $\boldsymbolβ=(β_0,\dots,β_{p-1})$ such that $δ$ is a Pisot number and $β_0,\ldots,β_{p-1}\in {\mathbb Q}(δ)$, 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 $A\subset {\mathbb Z}$ was introduced by Erdős et al. For our purposes, we use a generalized concept with $δ\in{\mathbb C}$ and $A\subset{\mathbb C}$ and study its topological properties.

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