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：令 $\alpha$ 和 $\beta$ 属于同一二次域。我们证明非齐次Beatty序列 $(\lfloor n \alpha + \beta \rfloor)_{n \geq 1}$ 是同步的，即存在一个有限自动机，该自动机并行读取 $n$ 和 $y$ 的Ostrowski表示，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，由此可得到一个新颖且简单的证明：这些序列在加法下的一阶逻辑理论是可判定的。该判定过程可在自由软件Walnut中轻松实现。作为应用，我们证明了对于每个 $r \geq 1$，集合 $\{ \lfloor n \alpha + \beta \rfloor \, : \, n \geq 1 \}$ 是否构成 $r$ 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，并显式刻画了Hildebrand等人的一个序列。