New lower bounds for cardinalities of higher dimensional difference sets and sumsets

TL;DR

This paper establishes new lower bounds for the sizes of difference sets and sumsets in higher-dimensional Euclidean spaces, providing sharp asymptotic results and answering longstanding questions in additive combinatorics.

Contribution

It introduces sharp lower bounds for difference sets in \\mathbb{R}^d, improving previous results and addressing open questions by Ruzsa and Stanchescu.

Findings

Proved a lower bound for |A-A| with a main term depending on dimension d.

Established new bounds for restricted difference sets.

Derived bounds for asymmetric sumsets in higher dimensions.

Abstract

Let $d \geq 4$ be a natural number and let $A$ be a finite, non-empty subset of $\mathbb{R}^d$ such that $A$ is not contained in a translate of a hyperplane. In this setting, we show that \[ |A-A| \geq \bigg(2d - 2 + \frac{1}{d-1} \bigg) |A| - O_{d}(|A|^{1- \delta}), \] for some absolute constant $\delta>0$ that only depends on $d$. This provides a sharp main term, consequently answering questions of Ruzsa and Stanchescu up to an $O_{d}(|A|^{1- \delta})$ error term. We also prove new lower bounds for restricted type difference sets and asymmetric sumsets in $\mathbb{R}^d$.