On a Kakeya-type problem II
Freiman, Gregory A. ; Stanchescu, Yonutz V.
Funct. Approx. Comment. Math., Tome 40 (2009) no. 1, p. 167-183 / Harvested from Project Euclid
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 $\theta \le 1.8$.
Publié le : 2009-12-15
Classification:  Inverse additive number theory,  Kakeya problem,  11P70,  11B75
@article{1261157808,
     author = {Freiman, Gregory A. and Stanchescu, Yonutz V.},
     title = {On a Kakeya-type problem II},
     journal = {Funct. Approx. Comment. Math.},
     volume = {40},
     number = {1},
     year = {2009},
     pages = { 167-183},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1261157808}
}
Freiman, Gregory A.; Stanchescu, Yonutz V. On a Kakeya-type problem II. Funct. Approx. Comment. Math., Tome 40 (2009) no. 1, pp.  167-183. http://gdmltest.u-ga.fr/item/1261157808/