Difference sets in higher dimensions
15th December 2020, 1:00 pm – 2:00 pm
Virtual (online) Zoom seminar; a link will be sent to the Bristol Combinatorics Seminar and Heilbronn Number Theory Seminar mailing lists, the week before the seminar.
Given a finite, non-empty set A of integers, it is well known that both the sumset A+A and the difference set A-A have cardinality at least 2|A| - 1. In 1973, Freiman generalised the sumset estimate for higher dimensional sets, and his result was optimal. In contrast to the sumset case, the analogous problem of finding sharp lower bounds for cardinalities of difference sets remains open in four and more dimensions. In this talk, we present a short survey of these results including our own improvement in this setting.