Beatty Sequences for a Quadratic Irrational: Decidability and Applications
<|reference_start|>Beatty Sequences for a Quadratic Irrational: Decidability and Applications: Let $\alpha$ and $\beta$ belong to the same quadratic field. We show that the inhomogeneous Beatty sequence $(\lfloor n \alpha + \beta \rfloor)_{n \geq 1}$ 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 $y = \lfloor n \alpha + \beta \rfloor$. 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 $r \geq 1$ it is decidable whether the set $\{ \lfloor n \alpha + \beta \rfloor \, : \, n \geq 1 \}$ 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.<|reference_end|>
@article{schaeffer2024beatty,
    title={Beatty Sequences for a Quadratic Irrational: Decidability and
  Applications},
    author={Luke Schaeffer, Jeffrey Shallit, Stefan Zorcic},
    journal={arXiv preprint arXiv:2402.08331},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.08331},
    primaryClass={math.NT cs.DM cs.FL math.CO math.LO}
}