Quantitative bounds for product of simplices in subsets of the unit cube

For each $1\leq i \le n$, let $k_i\geq 1$ and let $\Delta_i$ be a set of vertices of a non-degenerate simplex of $k_i+1$ 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 $\delta$, we show that there exists an interval $I=I(\Delta_1,\ldots, \Delta_n,A)$ of length at least $\exp(-\delta^{-C(\Delta_1,\ldots, \Delta_n)})$ such that for each $\lambda\in I$, the set $A$ contains $\Delta'_1\times \cdots \times \Delta'_n$, where each $\Delta_i'$ is an isometric copy of $\lambda\Delta_i$. 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.