Approximate quadratic varieties

Abstract

A classical result in additive combinatorics, which is a combination of Balog-Szemer´edi- Gowers theorem and Freiman’s theorem says that if a subset A of Fn p contains at least c|A|3 additive quadruples, then there exists a subspace V , comparable in size to A, such that |A ∩ V | ≥ Ωc(|A|). Motivated by the fact that higher order approximate algebraic structures play an important role in the theory of uniformity norms, it would be of interest to find higher order analogues of the mentioned result. In this talk, I will discuss a quadratic version of the approximate property in question, namely what it means for a set to be an approximate quadratic variety. I will also say something about structure of such a set.