Let $A$ be a finite set of integers and consider the lines determined by pairs of points of $P = \{(a, a^2) : a \in A\}$. The sum set of $A$ is the set of slopes of these lines and the product set of A is the set of $y$-intercepts. We know from the celebrated sum-product theorem of Erdős and Szemerédi that at least one of these sets is much larger than $|A|$. Geometrically, this observation can be phrased as follows: infinity cannot both be close to the minimum. Motivated by this observation, Yufei Zhao asked if this is a manifestation of a more general phenomenon.
The goal of the talk is to answer this in the affirmative.
Joint work with O. Roche-Newton, M. Rudnev and A. Warren.
Giorgis Petridis