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

For each $1\le i\le n$, let $k_i\ge 1$ and let ${\mathrm{\Delta }}_i$ 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 $I=I({\mathrm{\Delta }}_1,\dots ,{\mathrm{\Delta }}_n,A)$ of length at least $\exp (-{\delta }^{-C({\mathrm{\Delta }}_1,\dots ,{\mathrm{\Delta }}_n)})$ such that for each $\lambda \in I$, the set $A$ contains ${\mathrm{\Delta }}_1^{\mathrm{\prime }}\times \cdots \times {\mathrm{\Delta }}_n^{\mathrm{\prime }}$, where each ${\mathrm{\Delta }}_i^{\mathrm{\prime }}$ is an isometric copy of $\lambda {\mathrm{\Delta }}_i$. 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.