Expansions of the Group of Integers by Beatty Sequences
Ayhan G\"unayd{\i}n, Melissa \"Ozsahakyan

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 where is an irrational number and the elements of are of the form for some . 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 .
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