On Cartesian Products which Determine Few Distinct Distances

TL;DR

This paper investigates the structure of Cartesian product point sets with near-minimal distinct distances, showing they must have small difference sets, thus advancing understanding of geometric combinatorics.

Contribution

It proves that Cartesian product sets with near minimal distances have small difference sets, improving previous bounds and connecting geometric and additive combinatorics.

Findings

Sets of the form A×A with near minimal distances have small difference sets.

The result improves bounds established by Hanson and Roche-Newton.

Provides new insights into the structure of point sets with few distinct distances.

Abstract

Every set of points $\mathcal{P}$ determines $\Omega(|\mathcal{P}| / \log |\mathcal{P}|)$ distances. A close version of this was initially conjectured by Erd\H{o}s in 1946 and rather recently proved by Guth and Katz. We show that when near this lower bound, a point set $\mathcal{P}$ of the form $A \times A$ must satisfy $|A - A| \ll |A|^{2-\frac{2}{7}} \log^{\frac{1}{7}} |A|$. This improves recent results of Hanson and Roche-Newton.