A Szemerédi-type theorem for subsets of the unit cube

Durcik, Polona; Kovač, Vjekoslav

doi:10.2140/apde.2022.15.507

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A Szemerédi-type theorem for subsets of the unit cube

Polona Durcik and Vjekoslav Kovač

Vol. 15 (2022), No. 2, 507–549

DOI: 10.2140/apde.2022.15.507

Abstract

We investigate gaps of $n$ -term arithmetic progressions $x$ , $x + y$ , …, $x + (n - 1) y$ inside a positive-measure subset $A$ of the unit cube ${[0, 1]}^{d}$ . If lengths of their gaps $y$ are evaluated in the $ℓ^{p}$ -norm for any $p$ other than $1$ , $2$ , …, $n - 1$ , and $\infty$ , and if the dimension $d$ is large enough, then we show that the numbers $∥ y ∥_{ℓ^{p}}$ attain all values from an interval, the length of which depends only on $n$ , $p$ , $d$ , and the measure of $A$ . Known counterexamples prevent generalizations of this result to the remaining values of the exponent $p$ . We also give an explicit bound for the length of the aforementioned interval. The proof makes the bound depend on the currently available bounds in Szemerédi’s theorem on the integers, which are used as a black box. A key ingredient of the proof is power-type cancellation estimates for operators resembling the multilinear Hilbert transforms. As a byproduct of the approach we obtain a quantitative improvement of the corresponding (previously known) result for side lengths of $n$ -dimensional cubes with vertices lying in a positive-measure subset of ${({[0, 1]}^{2})}^{n}$ .

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Keywords

Euclidean Ramsey theory, arithmetic progression, density theorem, multilinear estimate, singular integral, oscillatory integral

Mathematical Subject Classification

Primary: 05D10, 42B20

Secondary: 11B30

Milestones

Received: 28 April 2020

Revised: 24 August 2020

Accepted: 27 October 2020

Published: 12 April 2022

Authors

Polona Durcik
Schmid College of Science and Technology Chapman University Orange, CA United States

Vjekoslav Kovač
Department of Mathematics Faculty of Science University of Zagreb Zagreb Croatia

Vol. 15, No. 2, 2022