Expansions of the Group of Integers by Beatty Sequences

TL;DR

This paper investigates the model theoretic properties of the integers expanded by a predicate for Beatty sequences generated by irrational numbers, establishing axiomatization, quantifier elimination, and definability results.

Contribution

It provides the first axiomatization and quantifier elimination for the structure $( ext{Z},+,P_r)$ with Beatty sequences, and characterizes definable sets.

Findings

Definable sets are not sparse unless finite

No reducts expand $( ext{Z},+)$ beyond the given structure

Axiomatization and quantifier elimination achieved

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 $(\Z,+)$.