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Stable sets and additive combinatorics

Daniel Palacin

Mittwoch, 13. November 2019 16:30Uhr

Given a subset A of a finite abelian group G, we denote by A+A the subset of elements of G which are sum of two elements of A. A fundamental question in additive combinatorics is to determine the structure of subsets A satisfying that A+A has size at most K times the size of A, where K is a fixed parameter. It is easy to verify that these subsets are translates of subgroups when K=1. Furthermore, for arbitrary K and for abelian groups of bounded exponent, a celebrated theorem of Ruzsa asserts that A is covered by a finite union of translates of subgroups, whose sizes are commensurable to the size of A. Improvements of this result have been subsequently obtained by many authors such as Green, Tao and Sanders, as well as Hrushovski who obtained an analogous result for non-abelian groups using model theoretic tools.

In this talk I shall present a model theoretic version of Ruzsa's theorem for subsets A satisfying suitable model theoretic conditions, such as stability. This is joint work Amador Martin-Pizarro and Julia Wolf.