An improvement on the number of simplices in F-q(d)

Let epsilon be a set of points in F-q(d). Bennett et al. (2016) proved that if \epsilon\ >> [GRAHICS] then epsilon determines a positive proportion of all k-simplices. In this paper, we give an improvement of this result in the case when epsilon is the Cartesian product of sets. Namely, we show that if kd epsilon is the Cartesian product of sets and [GRAHICS] = o(\epsilon), the number of congruence classes of k-simplices determined by epsilon is at least (1 - omicron(1)) [GRAPHICS] , and in some cases our result is sharp. (C) 2017 Elsevier B.V. All rights reserved.