Max Aires, Rutgers University

Abstract

Given sets A and B, let the sumset A+B={a+b : a in A, b in B}, and define A-B similarly. Given a set of real numbers, what can we say about the cardinalities of |A+A| and |A-A|? More generally, what about |A+A+A| and |A-A|, or other combinations of sums, differences, and scalars? While simple heuristic arguments suggested that |A+A|<=|A-A| or |A+A+A|>=|A-A| may always hold, we will show the existence of sets which contradict this. We will construct these sets using a clever induction argument, and revel in their existence. More generally, we shall discuss extensions to other problems in additive combinatorics involving cardinalities of dilates. Finally, we discuss briefly the possibility (or lack thereof) of applying this technology to the strange and exciting new world of selling back-alley trading cards.