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Difference sets in higher dimensions
Published online by Cambridge University Press: 14 December 2020
Abstract
Let d ≥ 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
$$\begin{equation*} |A-A| \geq (2d-2)|A| - O_d(|A|^{1- \delta}),\end{equation*}$$
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.
MSC classification
- Type
- Research Article
- Information
- Mathematical Proceedings of the Cambridge Philosophical Society , Volume 171 , Issue 3 , November 2021 , pp. 467 - 480
- Copyright
- © Cambridge Philosophical Society 2020
References
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