Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-27T05:49:42.921Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Balog, A. and Shakan, G.. Sum of dilates in vector spaces. North-West. Eur. J. Math. 1 (2015), 46–54.Google Scholar
Freiman, G. A., Heppes, A. and Uhrin, B.. A lower estimation for the cardinality of finite difference sets in $\mathbb{R}^n$ . Coll. Math. Soc. J. Bolyai, Budapest. 51 (1989), 125–139.Google Scholar
González Merino, B. and Henze, M.. A generalization of the discrete version of Minkowski’s fundamental theorem. J. Mathematika. 62 (3) (2016), 637–652.CrossRefGoogle Scholar
Mudgal, A.. Sums of linear transformations in higher dimensions. Q. J. Math. 70 (2019), no. 3, 965–984.CrossRefGoogle Scholar
Ruzsa, I. Z.. Sum of sets in several dimensions. Combinatorica. 14 (1994), no. 4, 485–490.CrossRefGoogle Scholar
Stanchescu, Y.. On finite difference sets. Acta Math. Hungar. 79 (1998), no. 1-2, 123–138.CrossRefGoogle Scholar
Stanchescu, Y.. An upper bound for d-dimensional difference sets. Combinatorica. 21 (2001), no. 4, 591–595.CrossRefGoogle Scholar
Tao, T. and Vu, V. H.. Additive combinatorics. Camb. Stud. Adv. Math. 105 (Cambridge University Press, Cambridge, 2006).Google Scholar
Uhrin, B.. Some estimations useful in geometry of numbers. Period. Math. Hungar. 11 (1980), 95–103.CrossRefGoogle Scholar
Uhrin, B., On a generalization of Minkowski’s convex body theorem. J. Number Theory 13 (1981), 192–209.CrossRefGoogle Scholar