Model-completeness and decidability of the additive structure of integers expanded with a function for a Beatty sequence

We introduce a model-complete theory which completely axiomatizes the structure $Z_{\alpha}=(Z, +, 0, 1, f)$ where $f : x \to \lfloor{\alpha} x \rfloor $ is a unary function with $\alpha$ a fixed transcendental number. When $\alpha$ is computable, our theory is recursively enumerable, and hence decidable as a result of completeness. Therefore, this result fits into the more general theme of adding traces of multiplication to integers without losing decidability.
Comment: In the current version, the abstract has undergone a minor modification