# Geometric ideas in Sum-Product theory

## March 21, 2017

The sum-product conjecture of Erdos-Szemeredi states that for any finite set $A$ of the reals $$\max(|A+A|, |AA|) \geq |A|^{2 - \varepsilon}.$$ Here $A+A$ (resp. $AA$) is the set of all pairwise sums (resp. products) of elements of $A$. We will go over some sum-product type estimates and see how ideas from geometry come into play when proving such estimates.