# The typical structure of sets with small sumset

Natasha Morrison

## Affiliation:

University of Cambridge

## Date:

Thu, 14/11/2019 - 11:00am

## Venue:

RC-4082, The Red Centre, UNSW

## Abstract:

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 $A,B \subset \mathbb{Z}$. 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 $k \ge (\log n)^4$: if $\omega \to \infty$ as $n \to \infty$ (arbitrarily slowly), then almost all sets $A \subset [n]$ with $|A| = k$ and $|A + A| \le \lambda k$ 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.

Natasha Morrison is a Research Fellow at Sidney Sussex College in Cambridge. She was a Postdoctoral Research Fellow at IMPA 2018-2019 after finishing her PhD at the University of Oxford under the supervision of Alex Scott.