Geometric progressions in vector sumsets over finite fields

摘要

Given two subsets A, B of the d dimensional vector space over the finite field F-q of q elements, we show that the sumsetA + B = {a + b vertical bar a is an element of A, b is an element of B}contains k distinct vectors of the form (v(1)lambda(j)(1),...,v(d)lambda(j)(d)), where j = 0,..., k - 1, with nonzero vectors (v(1),...,v(d)), (lambda(1),...,lambda(d)) is an element of F-q(d), whenever#A#B = cq(2d(1-1 /k)),for some constant c 0 which depends only on k. This improves the previous result of O. Ahamadi and I.E. Shparlinski (2007), which has the exponent 2d - 2/k. (C) 2020 Elsevier Inc. All rights reserved.