Functiones et Approximatio Commentarii Mathematici

On a Kakeya-type problem II

Gregory A. Freiman and Yonutz V. Stanchescu

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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 $\theta \le 1.8$.

Article information

Funct. Approx. Comment. Math., Volume 41, Number 2 (2009), 167-183.

First available in Project Euclid: 18 December 2009

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11P70: Inverse problems of additive number theory, including sumsets
Secondary: 11B75: Other combinatorial number theory

Inverse additive number theory Kakeya problem


Freiman, Gregory A.; Stanchescu, Yonutz V. On a Kakeya-type problem II. Funct. Approx. Comment. Math. 41 (2009), no. 2, 167--183. doi:10.7169/facm/1261157808.

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  • G.A. Freiman, Y.V. Stanchescu, On a Kakeya-type problem, Funct. Approx. Comment. Math. 37.1 (2007), 131--148.