Abstract

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.

Publication Details
Publication Type
Journal Article
Year of Publication
2024
URL
https://arxiv.org/abs/2402.08331
Journal
arXiv
Contributors
Groups
Date Published
02/2024