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Difference sets in higher dimensions

Part of: Sequences and sets

Published online by Cambridge University Press:  14 December 2020

AKSHAT MUDGAL
AKSHAT MUDGAL*
Affiliation:
Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN47907-2067, U.S.A. School of Mathematics, University of Bristol, Fry Building, Woodland Road, Bristol, BS8 1UG, UK. e-mails: am16393@bristol.ac.uk, amudgal@purdue.edu
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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

Primary: 11B13: Additive bases, including sumsets
Secondary: 11B30: Arithmetic combinatorics; higher degree uniformity
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

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