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

Abstract

We introduce a model-complete theory which completely axiomatizes the structure $Z_{\alpha }=〈Z,+,0,1,f〉$ where $f:x\mapsto \lfloor \alpha x\rfloor$ is a unary function with α a fixed transcendental number. Moreover, we show that decidability of $Z_{\alpha }$ is equivalent to computability of α. This result fits into the more general theme of adding traces of multiplication to integers without losing decidability.