The Additive Structure of Cartesian Products Spanning Few Distinct Distances

Abstract

Guth and Katz proved that any point set $\mathcal{P}$ in the plane determines $\Omega (|\mathcal{P}|/\log |\mathcal{P}|)$ distinct distances. We show that when near to this lower bound, a point set $\mathcal{P}$ of the form $A\times A$ must satisfy $|A-A|\ll |A{|}^{2-1/8}$.

This is the second iteration of this article, with minor typos corrected. It has been accepted for publication in Combinatorica

Cited by in corpus (1)

• On sets with few distinct distances