Difference sets in $\mathbb{R}^d$

TL;DR

This paper establishes a sharp lower bound on the size of difference sets for large finite subsets of Euclidean space, resolving a longstanding question in additive combinatorics.

Contribution

It proves the optimal lower bound for difference sets in d, confirming the bound's tightness and answering an open problem posed by Uhrin.

Findings

Established the best possible lower bound for difference sets in d.

Resolved an old open question in additive combinatorics.

Provided constructions showing the bound's optimality.

Abstract

Let $d \geq 2$ be a natural number. We show that $$|A-A| \geq \left(2d-2 + \frac{1}{d-1}\right)|A|-(2d^2-4d+3)$$ for any sufficiently large finite subset $A$ of $\mathbb{R}^d$ that is not contained in a translate of a hyperplane. By a construction of Stanchescu, this is best possible and thus resolves an old question first raised by Uhrin.