Natasha Morrison

(Department of Mathematics and Statistics, University of Victoria)

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.