Erdős and Szemerédi made the (still open) conjecture that for a finite set of natural numbers, A, either the sumset A+A, or else the productset AA, must be nearly as large as possible. A slightly different interpretation is that either A+A is large or log(A)+log(A) is large, where log(A) is the image of A under the (convex) logarithm function. This phenomenon is in fact more general, and extends to arbitrary convex functions f: if f has non-vanishing second derivative, then either A+A or else f(A)+f(A) is large. In recent work with Roche-Newton and Rudnev, we show that this growth persists when f has further non-vanishing derivatives.
This video is part of the Webinar in Additive Combinatorics series, and this is their YouTube channel.
