Submission date: 15 August 2017

Abstract:

The Pila-Wilkie theorem states that if a set is definable in an o-minimal structure R over the reals and contains `many' rational points, then it contains an infinite semialgebraic set. In this paper, we extend the theorem to two important model theoretic settings. Let R'=(R, P) be an expansion of R by a dense set P, which is either an elementary substructure of R, or it is independent. We prove that if a set is definable in R' and contains many rational points, then it is dense in an infinite semialgebraic set. Along the way, we introduce the notion of the 'algebraic trace part' of any set.

Mathematics Subject Classification:

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Full text arXiv 1708.03936: pdf, ps.