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Geometric progressions in sumsets over finite fields

Authors:	Omran Ahmadi Igor E Shparlinski

Institution:	(1) University of Waterloo, Waterloo, Ontario, Canada;(2) Macquarie University, Sydney, NSW, Australia

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

Keywords:	2000 Mathematics Subject Classification: 11B83 11T23 11T30
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