The previous talk in the series is here. The next talk in the series is here.
Let A be a set of real numbers. The sumset A+A is defined to be the set of all sums a+b such that a and b are elements of A, and the product set A.A is defined to be the set of all products ab. If A has size n (and A consists of positive reals), then the sumset and the product set each have size between 2n-1 and n(n+1)/2. The sum-product problem of Paul Erdos and Endre Szemerédi is the question of how small it is possible to make the larger of the two. They conjecture that for any α less than 2, if n is large enough then one or other of the two must be at least nα. This is a famous unsolved problem. Here I discuss a beautiful argument of József Solymosi that gives a lower bound of n4/3-c for any positive c. After the introduction and some definitions, we discuss Solymosi’s theorem and its proof.
This video was produced by Tim Gowers as part of his Part III course at the University of Cambridge. Printed notes for this course are available here.