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

Abstract

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

References in corpus (3)

• Finite Chains inside Thin Subsets of ${\mathbb{R}}^d$
• Nilfactors of R^m-actions and configurations in sets of positive upper density in R^m
• A Roth type theorem for dense subsets of ${\mathbb{R}}^d$

Cited by in corpus (8)

• A Szemerédi-type theorem for subsets of the unit cube
• Boxes, extended boxes, and sets of positive upper density in the Euclidean space
• Simplices in thin subsets of Euclidean spaces
• Density theorems for anisotropic point configurations
• The density of sets containing large similar copies of finite sets
• Coloring and density theorems for configurations of a given volume
• On Sets Containing an Affine Copy of Bounded Decreasing Sequences
• Large dilates of hypercube graphs in the plane