Expansions of the Group of Integers by Beatty Sequences

Abstract: We study the model theoretic structure $(\Z,+,P_r)$ where $r>1$ is an irrational number and the elements of $P_r$ are of the form $\floor{nr}$ for some $n\in\Z\setminus{0}$. We axiomatize of this structure and prove a quantifier elimination result. As a consequence, we get that definable subsets are not sparse unless they are finite. We also prove that there are no reducts of this structure expanding $(\Z,+)$.