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 $Δ_i$ 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 $δ$, we show that there exists an interval $I=I(Δ_1,\ldots, Δ_n,A)$ of length at least $\exp(-δ^{-C(Δ_1,\ldots, Δ_n)})$ such that for each $λ\in I$, the set $A$ contains $Δ'_1\times \cdots \times Δ'_n$, where each $Δ_i'$ is an isometric copy of $λΔ_i$. 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.

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