## Functiones et Approximatio Commentarii Mathematici

### On a Kakeya-type problem

#### 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 note we find the {\it exact value } of $R_s(k)$ in cases $s = 1$, $s = 2$ and $s = 3$. The case $s = 3$ appeared to be not simple. The structure of {\it extremal sets} led us to sets isomorphic to planar sets having a rather unexpected form of a perfect hexagon. The proof suggests the way of dealing with the general case $s \ge 4$.

#### Article information

Source
Funct. Approx. Comment. Math., Volume 37, Number 1 (2007), 131-148.

Dates
First available in Project Euclid: 18 December 2008

Permanent link to this document
https://projecteuclid.org/euclid.facm/1229618746

Digital Object Identifier
doi:10.7169/facm/1229618746

Mathematical Reviews number (MathSciNet)
MR2357314

Zentralblatt MATH identifier
1210.11106

#### Citation

Freiman, Gregory A.; Stanchescu, Yonutz V. On a Kakeya-type problem. Funct. Approx. Comment. Math. 37 (2007), no. 1, 131--148. doi:10.7169/facm/1229618746. https://projecteuclid.org/euclid.facm/1229618746

#### References

• J. Bourgain, On the dimension of Kakeya sets and related maximal inequalities, Geom. Funct. Anal. 9 (1999), no.2, 256--282.
• N.H. Katz and T. Tao, Bounds on arithmetic progressions and applications to the Kakeya conjecture, Mathematical Research Letters 6 (1999), 625--630.