Discrete sets definable in strong expansions of ordered Abelian groups

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).