On the size of $k$-fold sum and product sets of integers
Abstract
Description
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 $$ |A + ... + A| \geq |A|^b$$ or $$ |A \cdot ... \cdot A| \geq |A|^b$$ 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)
33 pages, no figures, submitted, J. Amer. Math. Soc. (Proxy submission)