Discrete sets definable in strong expansions of ordered Abelian groups

TL;DR

This paper investigates the structure of infinite discrete definable sets in strong, definably complete expansions of ordered Abelian groups, revealing finiteness properties and definability conditions related to the structure's burden.

Contribution

It establishes bounds on the complexity of discrete sets under burden constraints and characterizes definable discrete sets in densely ordered structures with burden 2.

Findings

Applying the difference operation n times yields a finite set for structures with burden at most n.

In densely ordered structures with burden 2, all definable discrete sets are definable in an extension of (R; <, +, Z).

Abstract

We study the structure of infinite discrete sets D definable in expansions of ordered Abelian groups whose theories are strong and definably complete, with particular emphasis on the set D' comprised of differences between successive elements. In particular, if the burden of the structure is at most n, then the result of applying the operation taking D to D' n times must be a finite set (Theorem 1.1). In the case when the structure is densely ordered and has burden 2, we show that any definable unary discrete set must be definable in some elementary extension of the structure (R; <, +, Z) (Theorem 1.3).