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.

翻译：设$\alpha$和$\beta$属于同一二次域。我们证明非齐次贝蒂序列$(\lfloor n \alpha + \beta \rfloor)_{n \geq 1}$是同步的，即存在一个有限自动机，该自动机以$n$和$y$的Ostrowski表示的并行输入，当且仅当$y = \lfloor n \alpha + \beta \rfloor$时接受。由于已知基于二次数的Ostrowski表示的加法关系是可计算的，其推论是：这些序列与加法的一阶逻辑理论可判定性的一个全新且相当简单的证明。该判定过程可轻松在自由软件Walnut中实现。作为应用，我们证明对于每个$r \geq 1$，集合$\{ \lfloor n \alpha + \beta \rfloor \, : \, n \geq 1 \}$是否形成$r$阶加法基（或渐近加法基）是可判定的。利用我们的技术，我们还解决了Reble和Kimberling的一些开放问题，并给出了Hildebrand等人一个序列的显式刻画。