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Density theorems for anisotropic point configurations

Part of: Harmonic analysis in several variables Classical measure theory Extremal combinatorics

Published online by Cambridge University Press:  03 May 2021

Vjekoslav Kovač
Vjekoslav Kovač*
Affiliation:
Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička cesta 30, 10000Zagreb, Croatia
*
e-mail: vjekovac@math.hr
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Abstract

Several results in the existing literature establish Euclidean density theorems of the following strong type. These results claim that every set of positive upper Banach density in the Euclidean space of an appropriate dimension contains isometric copies of all sufficiently large elements of a prescribed family of finite point configurations. So far, all results of this type discussed linear isotropic dilates of a fixed point configuration. In this paper, we initiate the study of analogous density theorems for families of point configurations generated by anisotropic dilations, i.e., families with power-type dependence on a single parameter interpreted as their size. More specifically, we prove nonisotropic power-type generalizations of a result by Bourgain on vertices of a simplex, a result by Lyall and Magyar on vertices of a rectangular box, and a result on distance trees, which is a particular case of the treatise of distance graphs by Lyall and Magyar. Another source of motivation for this paper is providing additional evidence for the versatility of the approach stemming from the work of Cook, Magyar, and Pramanik and its modification used recently by Durcik and the present author. Finally, yet another purpose of this paper is to single out anisotropic multilinear singular integral operators associated with the above combinatorial problems, as they are interesting on their own.

Keywords

Euclidean Ramsey theorypoint configurationdistance graphheat flowsingular integral

MSC classification

Primary: 05D10: Ramsey theory
Secondary: 28A75: Length, area, volume, other geometric measure theory 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Type
Article
Information
Canadian Journal of Mathematics , Volume 74 , Issue 5 , October 2022 , pp. 1244 - 1276
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Copyright
© Canadian Mathematical Society 2021

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Footnotes

This work is supported in part by the Croatian Science Foundation project UIP-2017-05-4129 (MUNHANAP) and in part by the Fulbright Scholar Program.

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