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We study the model theoretic structure $(\Z,+,P_r)$ where $r>1$ is anirrational number and the elements of $P_r$ are of the form $\floor{nr}$ forsome $n\in\Z\setminus\{0\}$. We axiomatize of this structure and prove aquantifier elimination result. As a consequence, we get that definable subsetsare not sparse unless they are finite. We also prove that there are no reductsof this structure expanding $(\Z,+)$.

Expansions of the Group of Integers by Beatty Sequences