Sumsets of sets of positive density in the integers
by
Endenicher Allee 60, Seminarraum 0.011
Mathezentrum
Abstract:
A central object of study in additive combinatorics is the sumset $A+B$ of two sets $A$ and $B$. Two of the basic questions one may ask are direct questions ("how large must $A+B$ be in terms of the sizes of $A$ and $B$?") and inverse questions ("if $A+B$ is small, what can be deduced about the structure of $A$ and $B$?"). When $A$ and $B$ are infinite subsets of the integers with size quantified by natural density $d(·)$, Kneser (1953) proved the direct theorem that $d(A+B) \ge d(A) + d(B)$ unless $A$ and $B$ have certain modular obstructions. Erdős and Graham (1980) asked for a corresponding inverse theorem classifying sets with $d(A+B) = d(A) + d(B)$. In this talk, we will present a new result characterizing the pairs of sets satisfying $d(A+B) = d(A) + d(B)$ in the absence of modular obstructions. We will also highlight some of the key Fourier analytic and ergodic theoretic aspects of the proof. This talk is based on joint work with Florian K. Richter.