# Patterns in sets of positive density in trees and affine buildings

## TL;DR

This paper extends results on distance patterns from Euclidean spaces to homogeneous trees and affine buildings, and constructs a counterexample in a non-homogeneous tree showing limitations of such patterns.

## Contribution

It proves an analogue of Bourgain's result for homogeneous trees and affine buildings, and provides a counterexample in a non-homogeneous tree.

## Key findings

- Homogeneous trees and affine buildings exhibit similar distance pattern properties to Euclidean spaces.
- A non-homogeneous tree with positive Hausdorff dimension can have subsets lacking large even distances.
- The results highlight differences between homogeneous and non-homogeneous structures in geometric combinatorics.

## 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.