# Natasha Morrison

## The Typical Structure of Sets with small sumset

Date Friday, November 20, 2020

One of the central objects of interest in additive combinatorics is the sumset $$A+B= \{a+b:\ a \in A, b \in B \}$$ of two sets A,B of integers.

Our main theorem, which improves results of Green and Morris, and of Mazur, implies that the following holds for every fixed $$\lambda > 2$$ and every $$k>(\log n)^4$$: if $$\omega$$ goes to infinity as n goes to infinity (arbitrarily slowly), then almost all sets $$A$$ of $$[n]$$ with $$\vert A\vert = k$$ and $$\vert A + A \vert < \lambda k$$ are contained in an arithmetic progression of length $$\lambda k/2 + \omega$$.

This is joint work with Marcelo Campos, Mauricio Collares, Rob Morris and Victor Souza.