A Short Proof of a Near-Optimal Cardinality Estimate for the Product of a Sum Set

Abstract

In this note it is established that, for any finite set A of real numbers, there exist two elements a, b from A such that |(a + A)(b + A)| > c|A|^2 / log |A|, where c is some positive constant. In particular, it follows that |(A + A)(A + A)| > c|A|^2 / log |A|. The latter inequality had in fact already been established in an earlier work of the author and Rudnev, which built upon the recent developments of Guth and Katz in their work on the Erdös distinct distance problem. Here, we do not use those relatively deep methods, and instead we need just a single application of the Szemerédi-Trotter Theorem. The result is also qualitatively stronger than the corresponding sum-product estimate from the paper of the author and Rudnev, since the set (a + A)(b + A) is defined by only two variables, rather than four. One can view this as a solution for the pinned distance problem, under an alternative notion of distance, in the special case when the point set is a direct product A x A. Another advantage of this more elementary approach is that these results can now be extended for the first time to the case when A is a set of complex numbers.