On distance sets of large sets of integer points
type: Item
instance of: scholarly article
title: On distance sets of large sets of integer points
zbMATH Open document ID: 1153.11006
author: Ákos Magyar
published in: Israel Journal of Mathematics
publication date: 2008-09-03T00:00:00Z
review text: Let \(\mathbb Z\) denote the set of integers and let \(n\) be a positive integer. We say that a set \(A\subseteq \mathbb Z^n\) has \textit{upper density at least} \(\varepsilon\) (where \(\varepsilon\) is a real number, \(0<\varepsilon\leq1),\) if there is a sequence of \(n\)-dimensional cubes \(B_{R_j}\) , \(j=1,2,\dots,\) of sizes \(R_j\rightarrow\infty,\) not necessarily centered at the origin, such that, for all \(j,\) \[ |A\cap B_{R_j}|\geq\varepsilon {R_j}^n. \] The main result of the paper is the following theorem. Let \(n\geq5,\) \(1\geq\varepsilon>0\) and \(A\subseteq \mathbb Z^n\) having upper density at least \(\varepsilon.\) Then there exist a positive integer \(Q_\varepsilon,\) depending only on \(\varepsilon,\) and a positive integer \(\Lambda_A\) depending on the set \(A,\) such that, for every integer \(\lambda\geq\Lambda_A,\) \(\lambda Q_\varepsilon^2\) belongs to the set \(\{|m-\ell|^2~;~m\in A,\ell\in A\}.\) Here \(m=(m_1,\dots,m_n),\ell= (\ell_1,\dots,\ell_n),\) and \(|m-\ell|\) denotes the distance \(\sum_{i=1}^n|m_i-\ell_i|.\) The author explains why such a result does not hold for \(n\leq 3\) and leaves open the case \(n=4.\) The work is motivated by the papers [\textit{H. Furstenberg, Y. Katznelson} and \textit{B. Weiss}, Algorithms Comb. 5, 184--198 (1990; Zbl 0738.28013) and \textit{J. Bourgain}, Isr. J. Math. 54, 307--316 (1986; Zbl 0609.10043)].
reviewed by: Georges Grekos
Mathematics Subject Classification ID: 11B05
zbMATH DE Number: 5320440
zbMATH Keywords: density
zbMATH Keywords: integer points
MaRDI profile type: Publication
full work available at URL: s11856-008-0028-z
OpenAlex ID: W1973774469
cites work: A Szemerédi type theorem for sets of positive density in \(\mathbb R^k\)
cites work: Q3972759
cites work: Distance sets of well-distributed planar point sets
cites work: Distance sets corresponding to convex bodies
cites work: Discrete analogues in harmonic analysis: spherical averages
DOI: 10.1007/S11856-008-0028-Z
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description: scientific article; zbMATH DE number 5320440