On the size of $k$-fold sum and product sets of integers

Abstract

We prove the following theorem: for all positive integers $b$ there exists a positive integer $k$, such that for every finite set $A$ of integers with cardinality $|A|>1$, we have either $$|A+...+A|\ge |A{|}^b$$ or $$|A\cdot ...\cdot A|\ge |A{|}^b$$ where $A+...+A$ and $A\cdot ...\cdot A$ are the collections of $k$-fold sums and products of elements of $A$ respectively. This is progress towards a conjecture of Erdös and Szemerédi on sum and product sets.

33 pages, no figures, submitted, J. Amer. Math. Soc. (Proxy submission)