Quantitative Bounds for Products of Simplices in Subsets of the Unit Cube

Authors

Polona Durcik, Chapman UniversityFollow
Mario Stipčić, Chapman UniversityFollow

Document Type

Article

Publication Date

2-3-2025

Abstract

For each 1 ≤ i ≤ n, let k_i ≥ 1 and let Δ_i be a set of vertices of a non-degenerate simplex of k_i + 1 points in R^ki+1. If A ⊆ [0, 1]^k1+1 × ·· ·×[0, 1]^kn+1 is a Lebesgue measurable set of measure at least δ, we show that there exists an interval I = I(Δ₁, . . . ,Δ_n,A) of length at least exp(−δ^{−C(Δ1,...,Δn)}) such that for each λ ∈ I, the set A contains Δ'₁ × ·· ·×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.

Comments

This article was originally published in Israel Journal of Mathematics in 2025. https://doi.org/10.1007/s11856-024-2710-1

Recommended Citation

Durcik, P., Stipčić, M. Quantitative bounds for products of simplices in subsets of the unit cube. Isr. J. Math. (2025). https://doi.org/10.1007/s11856-024-2710-1

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1

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