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arXiv · 2007.11526

Difference sets in higher dimensions

Akshat Mudgal
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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 \[ |A-A| \geq (2d-2)|A| - O_d(|A|^{1- δ}),\] where $δ>0$ 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.

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Akshat Mudgal. 2020-07-22. Difference sets in higher dimensions. https://doi.org/10.1017/s0305004120000298

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