Submission date: 26 January 2017

Abstract:

We consider the question of when sets definable in first-order expansions of groups contain the product of two infinite sets (we refer to this as the “productset property”). We show that the productset property holds for any definable subset A of an expansion of a discrete amenable group such that A has positive Banach density and the formula x &midxdot; y ∈ A is stable. For arbitrary first-order expansions of groups, we consider a “1-sided” version of the productset property, which is characterized in various ways using coheir independence. For stable groups, the productset property is equivalent to this 1-sided version, and behaves as notion of largeness for definable sets, which can be characterized by a natural weakening of model-theoretic genericity.

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Full text arXiv 1701.07791: pdf, ps.