Patterns in sets of positive density in trees and affine buildings

2019/07/12 by M. Björklund, Michael Björklund, A. Fish +6
Mathematics · #Advanced Topology and Set Theory #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals #math.CO #msc:05D10

paper · pdf · doi:10.48550/arxiv.1907.05825

arxiv created 2019/07/12 · arxiv updated 2019/07/15

Abstract

We prove an analogue for homogeneous trees and certain affine buildings of a result of Bourgain on pinned distances in sets of positive density in Euclidean spaces. Furthermore, we construct an example of a non-homogeneous tree with positive Hausdorff dimension, and a subset with positive density thereof, in which not all sufficiently large (even) distances are realised.

Related

• Pinned distances and density theorems in \mathbb R^d
• Simplices and sets of positive upper density in ℝ^d
• Distances and Trees in Dense Subsets of ℤ^d
• Nonempty interior of pinned distance and tree sets
• Trees and Homogeneous LOTS