Somewhere between Freiman's theorem and the Polynomial Freiman-Ruzsa conjecture
Seminar Room 1, Newton Institute
Freiman's theorem describes the structure of sets of integers having small doubling which enjoys numerous applications in additive combinatorics. While the result is qualitatively comprehensive, a truer picture is suggested by the Polynomial Freiman-Ruzsa conjecture, and in this talk we shall discuss how a remarkable new technique of Croot and Sisask can be used in conjunction with a trick of Lopez and Ross to provide much clearer quantitative information. We shall focus on the model finite field setting where there are few prerequisites and the details are the cleanest.