Naslov
Quantitative bounds for products of simplices in subsets of the unit cube

Autori
Durcik, Polona ; Stipčić, Mario

Vrsta, podvrsta
Radovi u časopisima, znanstveni

Izvornik
Israel journal of mathematics (2023)

Status rada
Prihvaćen

Ključne riječi
Euclidean Ramsey theory, point configuration, singular integral

Sažetak
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 Rki+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, …, Δn, 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 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.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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