Geometric progressions in sumsets over finite fields

Given two sets A, B í \Bbb F_q^d{\cal A}, {\cal B}\subseteq {\Bbb F}_q^d , the set of d dimensional vectors over the finite field \Bbb F_q{\Bbb F}_q with q elements, we show that the sumset A+B = {a+b | a ? A, b ? B}{\cal A}+{\cal B} = \{{\bf a}+{\bf b}\ \vert\ {\bf a} \in {\cal A}, {\bf b} \in {\cal B}\} contains a geometric progression of length k of the form vΛ ^j , where j = 0,…, k − 1, with a nonzero vector v ? \Bbb F_q^d{\bf v} \in {\Bbb F}_q^d and a nonsingular d × d matrix Λ whenever # A # B 3 20 q^2d-2/k\# {\cal A} \# {\cal B} \ge 20 q^{2d-2/k} . We also consider some modifications of this problem including the question of the existence of elements of sumsets on algebraic varieties.