## Variations on the sum-product problem

#### Oliver Roche-NewtonUniversity of ReadingMathematics

The Erdos-Szemerédi sum-product conjecture, roughly speaking, states that for a given set of integers, either its sum set or product set must be almost as large as possible. Speaking even more vaguely, this conjecture claims that a set cannot be highly structured in both an additive and multiplicative sense, and this talk aims to introduce some results which make this notion more precise. In particular, the main idea is to show that a set which is defined by a combination of additive and multiplicative operations must be large; these sets are sometimes known as expanders. Some extremal versions of these questions will also be discussed.

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