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December 2009 On a Kakeya-type problem II
Gregory A. Freiman, Yonutz V. Stanchescu
Funct. Approx. Comment. Math. 41(2): 167-183 (December 2009). DOI: 10.7169/facm/1261157808

Abstract

Let $A$ be a finite subset of an abelian group $G$. For every element $b_i$ of the sumset $2A=\{b_0, b_1, ...,b_{|2A|-1}\}$ we denote by $D_i=\{a-a': a, a' \in A; a+a'=b_i\}$ and $r_i=|\{(a,a'): a+a'=b_i; a, a' \in A \}|$. After an eventual reordering of $2A$, we may assume that $r_0\geq r_1 \geq ...\geq r_{|2A|-1}.$ For every $1 \le s \le |2A|$ we define $R_s(A)=|D_0\cup D_1\cup...\cup D_{s-1}|$ and $R_s(k)=\max \{R_s(A): A\subseteq G, |A| =k\}.$ Bourgain and Katz and Tao obtained an estimate of $R_s(k)$ assuming $s$ being of order $k$. In this paper we describe the {\it structure} of $A$ assuming that\break $G=\mathbb{Z}^2, s=3$ and $R_3(A)$ is close to its maximal value, i.e. $R_3(A) = 3k-\theta \sqrt{k}$, with $\theta \le 1.8$.

Citation

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Gregory A. Freiman. Yonutz V. Stanchescu. "On a Kakeya-type problem II." Funct. Approx. Comment. Math. 41 (2) 167 - 183, December 2009. https://doi.org/10.7169/facm/1261157808

Information

Published: December 2009
First available in Project Euclid: 18 December 2009

zbMATH: 1211.11110
MathSciNet: MR2590332
Digital Object Identifier: 10.7169/facm/1261157808

Subjects:
Primary: 11P70
Secondary: 11B75

Rights: Copyright © 2009 Adam Mickiewicz University

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Vol.41 • No. 2 • December 2009
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