Distance graphs and sets of positive upper density in ℝd

Lyall, Neil; Magyar, Ákos

doi:10.2140/apde.2020.13.685

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Distance graphs and sets of positive upper density in $\mathbb{R}^d$

Neil Lyall and Ákos Magyar

Vol. 13 (2020), No. 3, 685–700

DOI: 10.2140/apde.2020.13.685

Abstract

We present a refinement and sharp extension of a result of Bourgain on finding configurations of $k + 1$ points in general position in measurable subset of $ℝ^{d}$ of positive upper density whenever $d \geq k + 1$ to all proper $k$ -degenerate distance graphs.

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Keywords

distance graphs, uniformity norms, geometric Ramsey theory

Mathematical Subject Classification 2010

Primary: 11B30

Milestones

Received: 23 March 2018

Revised: 11 January 2019

Accepted: 13 March 2019

Published: 15 April 2020

Authors

Neil Lyall
Department of Mathematics The University of Georgia Athens, GA United States

Ákos Magyar
Department of Mathematics The University of Georgia Athens, GA United States

Vol. 13, No. 3, 2020