Sum-product Inequalities and Combinatorial Problems on Sumsets

Dissertation Defense
Friday, July 17, 2015 - 2:00pm for 1 hour (actually 50 minutes)
Skiles 006
Albert Bush – School of Mathematics, Georgia Tech
Albert Bush
The thesis investigates a version of the sum-product inequality studied by Chang in which one tries to prove the h-fold sumset is large under the assumption that the 2-fold product set is small. Previous bounds were logarithmic in the exponent, and we prove the first super-logarithmic bound. We will also discuss a new technique inspired by convex geometry to find an order-preserving Freiman 2-isomorphism between a set with small doubling and a small interval. Time permitting, we will discuss some combinatorial applications of this result.