Visualizing the sum-product conjecture

Abstract: The Erdos sum-product conjecture states that, for every $\epsilon>0$, there is $k_0$ such that if $A$ is any finite set of positive integers with $|A|>k_0$, then $|(A+A)\cup(AA)| > |A|^{2-\epsilon}$. In other words, for sufficiently large sets either the sumset or the product set will be nearly as large as conceivable. We survey progress on this conjecture, and provide a visual representation of progress and counterexamples. There will be a few beautiful proofs (not the speaker's), several interesting examples, and scores of striking pictures.{

commutative algebracombinatoricsnumber theory

Audience: researchers in the topic

New York Number Theory Seminar