(Department of Mathematics and Statistics, University of Victoria)
The Typical Structure of Sets with small sumset
| Date | Friday, November 20, 2020 |
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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.