Beatty Sequences for a Quadratic Irrational: Decidability and Applications
This work addresses foundational problems in number theory and automata theory, providing new decidability results and applications, though it is incremental in building on known techniques.
The paper tackles the decidability of inhomogeneous Beatty sequences for quadratic irrationals by showing they are synchronized via finite automata, leading to a simple proof that their first-order logical theory with addition is decidable, and applies this to solve open problems and characterize sequences.
Let $α$ and $β$ belong to the same quadratic field. We show that the inhomogeneous Beatty sequence $(\lfloor n α+ β\rfloor)_{n \geq 1}$ 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 $y = \lfloor n α+ β\rfloor$. 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 $\{ \lfloor n α+ β\rfloor \, : \, n \geq 1 \}$ 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.