A Euclidean Fourier-analytic approach to vertical projections in the Heisenberg group

TL;DR

This paper improves the lower bounds for Hausdorff dimension under vertical projections in the Heisenberg group using Euclidean Fourier analysis, addressing a question posed by F"assler and Hovila.

Contribution

It introduces a novel Fourier-analytic method employing Bessel functions to enhance dimension bounds in the Heisenberg group setting.

Findings

Established a better a.e. lower bound for Hausdorff dimension

Answered an open question by F"assler and Hovila

Applied Euclidean Fourier techniques to sub-Riemannian geometry

Abstract

An improved a.e. lower bound is given for Hausdorff dimension under vertical projections in the first Heisenberg group, with respect to the Carnot-Carath\'eodory metric. This improves the known lower bound, and answers a question of F\"assler and Hovila. The approach uses the Euclidean Fourier transform, Basset's integral formula, and modified Bessel functions of the second kind.