An improvement on the number of simplices in Fqd

Let E be a set of points in Fqd. Bennett et al. (2016) proved that if |E|≫qd−d−1k+1 then E determines a positive proportion of all k-simplices. In this paper, we give an improvement of this result in the case when E is the Cartesian product of sets. Namely, we show that if E is the Cartesian product of sets and qkdk+1−1∕d=o(|E|), the number of congruence classes of k-simplices determined by E is at least (1−o(1))qk+12, and in some cases our result is sharp.