On a Kakeya-type Problem

Let A be a finite subset of an abelian group G . For every element bi of the sumset 2A={b0, b1, ..., b|2A|−1} we denote by Di={a − a′ : a, a′∈A; a+a′=bi} and ri=|{(a, a′) : a+a′=bi; a, a′∈A}| . After an eventual reordering of 2A , we may assume that r0>r1>...>r|2A|−1. For every 16s6|2A| we define Rs(A) = |D0∪D1∪...∪Ds−1| and Rs(k)= max{Rs(A) : A⊆G, |A|=k}. Bourgain and Katz and Tao obtained an estimate of Rs(k) assuming s being of order k . In this note we find the exact value of Rs(k) in cases s=1 , s=2 and s=3 . The case s=3 appeared to be not simple. The structure of 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>4 .