Patterns in sets of positive density in trees and affine buildings

M. Björklund, A. Fish, J. Parkinson

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.

Keywords: Density, Ramsey theory, trees, buildings.

AMS Subject Classification: Primary 05D10.