Vol. 15, No. 2, 2022

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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

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 p-norm for any p other than 1, 2, …, n 1, and , and if the dimension d is large enough, then we show that the numbers yp 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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Euclidean Ramsey theory, arithmetic progression, density theorem, multilinear estimate, singular integral, oscillatory integral
Mathematical Subject Classification
Primary: 05D10, 42B20
Secondary: 11B30
Received: 28 April 2020
Revised: 24 August 2020
Accepted: 27 October 2020
Published: 12 April 2022
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