Difference sets in higher dimensions

TL;DR

This paper proves a lower bound on the size of difference sets in higher-dimensional Euclidean spaces for sets not contained in a hyperplane, advancing understanding of additive combinatorics in multiple dimensions.

Contribution

It establishes a new lower bound on difference sets in -dimensional space, improving previous results and progressing toward a longstanding conjecture.

Findings

Lower bound |A| - O_d(|A|^{1-\u03b4}) for difference sets in D

Improves upon earlier results by Freiman, Heppes, and Uhrin

Makes progress towards a conjecture of Stanchescu

Abstract

Let $d \geq 3$ be a natural number. We show that for all finite, non-empty sets $A \subseteq \mathbb{R}^d$ that are not contained in a translate of a hyperplane, we have \[ |A-A| \geq (2d-2)|A| - O_d(|A|^{1- \delta}),\] where $\delta >0$ is an absolute constant only depending on $d$. This improves upon an earlier result of Freiman, Heppes and Uhrin, and makes progress towards a conjecture of Stanchescu.