On a Kakeya-type problem

Let A be a flnite subset of an abelian group G. For every element b_i of the sumset 2A=b₀, b₁,.., b₂A-1 we denote by D_i=a-a′. a, a′ϵA, a+a′=b_i and r_i=(a, a′). a+a′=b_i, a, a′ϵA. After an eventual reordering of 2A, we may assume that r0≥r₁≥..≥r_2A-1. For every 1≤s≤2A we deflne Rs(A) = D₀[D₁[..[D_s-1 and Rs(k)= maxR_s(A). AG, A=kg. Bourgain and Katz and Tao obtained an estimate of R_s(k) assuming s being of order k. In this note we flnd the 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 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.