For each 1 ≤ i ≤ n, let ki ≥ 1 and let Δi be a set of vertices of a non-degenerate simplex of ki + 1 points in \({\mathbb R}^{{k_{i}}+1}\) . If \(A \subseteq [0,1]^{k_{1}+1} \times \cdots \times [0,1]^{{k_{n}}+1}\) is a Lebesgue measurable set of measure at least δ, we show that there exists an interval I = I(Δ1,…,Δn, A) of length at least \(\exp(-\delta^{{-C}(\Delta_{1},\ldots,\Delta_{n})})\) such that for each λ ∈ I, the set A contains \({\Delta^{\prime}_{1}} \times \cdots \times {\Delta^{\prime}_{n}}\) , where each \({\Delta^{\prime}_{i}}\) is an isometric copy of λΔi.This is a quantitative improvement of a result by Lyall and Magyar. Our proof relies on harmonic analysis. The main ingredient in the proof are cancellation estimates for forms similar to multilinear singular integrals associated with n-partite n-regular hypergraphs.
