The Additive Structure of Cartesian Products Spanning Few Distinct Distances

TL;DR

This paper investigates the structure of point sets with near-minimal distinct distances, showing that such sets of the form A×A have additive properties with differences bounded by approximately |A|^{15/8}.

Contribution

It establishes a new link between minimal distance configurations and additive combinatorics for Cartesian product point sets.

Findings

Point sets close to the minimal number of distinct distances have restricted additive structure.

Sets of the form A×A with near minimal distances satisfy |A-A| ≪ |A|^{15/8}.

Provides bounds connecting geometric and additive properties of point sets.

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 $|A-A|\ll |A|^{2-1/8}$.