# Finite trees inside thin subsets of ${\Bbb R}^d$

## TL;DR

This paper extends previous results on the presence of geometric configurations within large Hausdorff dimension sets in Euclidean space, demonstrating that such sets contain complex tree structures with specified properties.

## Contribution

It generalizes earlier work by Bennett, Iosevich, and Taylor to include arbitrary tree configurations within sets of sufficiently large Hausdorff dimension.

## Key findings

- Sets with Hausdorff dimension > (d+1)/2 contain arbitrary tree configurations.
- The result applies to all dimensions d ≥ 2.
- Generalizes chain results to more complex tree structures.

## Abstract

Bennett, Iosevich and Taylor proved that compact subsets of ${\Bbb R}^d$, $d \ge 2$, of Hausdorff dimensions greater than $\frac{d+1}{2}$ contain chains of arbitrary length with gaps in a non-trivial interval. In this paper we generalize this result to arbitrary tree configurations.