A Fourier Coefficients Approach to Hausdorff Dimension in the Heisenberg Group

TL;DR

This paper explores how Fourier analysis on the Heisenberg group can be used to understand the geometric properties of measures, particularly in relation to their densities and Hausdorff dimension.

Contribution

It introduces a novel approach linking Fourier coefficients to measure densities and provides a framework for using Fourier methods to compute Hausdorff dimension in the Heisenberg group.

Findings

Fourier transform square integrability implies measure has a square integrable density.

Integrable Fourier transform implies measure has a continuous density.

Fourier methods can be used to compute Hausdorff dimension of sets.

Abstract

This paper establishes connections between the group-Fourier transform and the geometry of measures in the Heisenberg group. Firstly, it is shown that if the Fourier transform of a compactly supported, finite, Radon measure is square integrable, then the measure must have a square integrable density. If it's Fourier transform is integrable, the the measure must have a continuous density. In addition, an alternative formulation of the Fourier transform on the Heisenberg group is used to show that energies of measures can be computed via integrals on an appropriate frequency space. This in turns opens the possibility of using Fourier methods in the computation of Hausdorff dimension of sets.