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Let E be a set of points in F q d . Bennett et al. (2016) proved that if | E | ≫ q d − d − 1 k + 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 q k d k + 1 − 1 ∕ d = o ...