# A comparison of Euclidean and Heisenberg Hausdorff measures

## TL;DR

This paper investigates the geometric properties of sets in the first Heisenberg group that have extremal relationships between their Euclidean and Heisenberg Hausdorff dimensions and measures, revealing their horizontal or vertical nature.

## Contribution

It establishes new geometric characterizations of sets with extremal Hausdorff dimensions in the Heisenberg group, demonstrating their horizontal or vertical structure.

## Key findings

- Sets with minimal Heisenberg Hausdorff dimension are horizontal.
- Sets with maximal Heisenberg Hausdorff dimension are vertical.
- Results are sharp, supported by concrete examples.

## Abstract

We prove some geometric properties of sets in the first Heisenberg group whose Heisenberg Hausdorff dimension is the minimal or maximal possible in relation to their Euclidean one and the corresponding Hausdorff measures are positive and finite. In the first case we show that these sets must be in a sense horizontal and in the second case vertical. We show the sharpness of our results with some examples.