Vol. 13, No. 3, 2020

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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 d of positive upper density whenever d k + 1 to all proper k-degenerate distance graphs.

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