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27.11.2012 (Tuesday)

Sums, Restricted Sums and Differences

Regular Seminar David Penman (Essex)

16:00 City U.
room CG04

Given a (nonempty) set $A$ of integers, two of the most obvious things to do with it are to form the sumset $A+A=\{a+b:\,a,b\in A\}$ and the difference set $A-A=\{a-b:\,a,b\in A\}$. One might also wish to consider the restricted sumset $A\hat{+}A=\{a+b:\,a,b\in A,\,a\neq b\}$. One can then ask various obvious questions about the relationships between the sizes of various of these sets and what this implies about structure, and I shall discuss some known results on this, including generalisations to more general contexts, e.g. in group theory. An intuition one might have is that the sumset/restricted sumset will be smaller than the difference set as addition is commutative but subtraction isn't: I shall survey various known results showing that this intuition is non-trivially wrong. At the end I shall discuss some recent constructions of sets $A$ which give new record large values of $\log(|A+A|)/\log(|A-A|)$. The original part of the talk is based on joint work with my research student Matthew Wells.