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 $a,b\in A$ such that $$|(a+A)(b+A)|\gg \frac{|A{|}^2}{\log |A|}.$$ In particular, it follows that $|(A+A)(A+A)|\gg \frac{|A{|}^2}{\log |A|}$. The latter inequality had in fact already been established in an earlier work of the author and Rudnev (arXiv:1203.6237), which built upon the recent developments of Guth and Katz (arXiv:1011.4105) in their work on the Erdős 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 (arXiv:1203.6237), 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 $A\times A$. Another advantage of this more elementary approach is that these results can now be extended for the first time to the case when $A\subset \mathbb{C}$.

To appear in Proceedings of SoCG 2015

References in corpus (2)

• On the Minkowski distances and products of sum sets
• New results for the growth of sets of real numbers

Cited by in corpus (1)

• On tripling constant of multiplicative subgroups