Beatty Sequences for a Quadratic Irrational: Decidability and Applications

TitleBeatty Sequences for a Quadratic Irrational: Decidability and Applications
Publication TypeJournal Article
Year of Publication2024
AuthorsSchaeffer, L, Shallit, J, Zorcic, S
Date Published2/13/2024

Let α and β belong to the same quadratic field. We show that the inhomogeneous Beatty sequence (⌊nα+β⌋)n≥1 is synchronized, in the sense that there is a finite automaton that takes as input the Ostrowski representations of n and y in parallel, and accepts if and only if y=⌊nα+β⌋. Since it is already known that the addition relation is computable for Ostrowski representations based on a quadratic number, a consequence is a new and rather simple proof that the first-order logical theory of these sequences with addition is decidable. The decision procedure is easily implemented in the free software Walnut.  As an application, we show that for each r≥1 it is decidable whether the set {⌊nα+β⌋:n≥1} forms an additive basis (or asymptotic additive basis) of order r. Using our techniques, we also solve some open problems of Reble and Kimberling, and give an explicit characterization of a sequence of Hildebrand et al.