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DISCRETE SETS DEFINABLE IN STRONG EXPANSIONS OF ORDERED ABELIAN GROUPS
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- Journal:
- The Journal of Symbolic Logic , First View
- Published online by Cambridge University Press:
- 13 December 2024, pp. 1-37
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We study the structure of infinite discrete sets D definable in expansions of ordered Abelian groups whose theories are strong and definably complete, with a 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 
$D \mapsto D'\ n$ 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 
$\langle \mathbb{R}; <, +, \mathbb{Z} \rangle $ (Theorem 1.3).
