Spectrum, algebraicity and normalization in alternate bases

Abstract

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 = p−1 i=0 βi 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.

On Periodic Alternate Base Expansions

“…We start by proving two equivalent conditions on x ∈ [0, 1) in order to belong to Per(β), allowing us to use the remainders of the greedy algorithms only once out of p steps. Then we obtain a generalization of a result from [6]. This generalization represents a crucial argument in the proof of our main result.…”

“…As a direct consequence of Theorem 1, we reobtain the above-mentioned result from [6] generalizing the fact that all Pisot numbers are Parry numbers. As a second corollary, we obtain the following property of Pisot numbers.…”

“…, β p−1 all belong to the extension field Q(β) is trivially satisfied whenever p = 1, that is, in the original case of Rényi's expansions. This condition already appeared in [6]. In that work, the aim was to obtain algebraic descriptions of alternate bases β = (β 0 , .…”

“…Our proof of Theorem 1 is based on algebraic tools such as the alternate base spectrum defined in [6] as a generalization of the β-spectrum originally introduced by Erdős, Joó and Komornik [7]. In the reduced case of one real base, we obtain a proof that is much shorter than Schmidt's original one from [12].…”

“…It was then shown in [6] that a necessary condition for the β-shift to be sofic is that the product β = p−1 i=0 β i is an algebraic integer and all of the bases β 0 , . .…”