Packing dimension of vertical projections in the Heisenberg group

TL;DR

This paper investigates the packing and Hausdorff dimensions of vertical projections of sets in the Heisenberg group, establishing lower bounds and improving known results using a variable coefficient local smoothing inequality.

Contribution

It provides new lower bounds for the packing and Hausdorff dimensions of projections in the Heisenberg group, extending previous results for sets with dimensions between 2 and 3.

Findings

Packing dimensions of projections are almost surely not less than the set's dimension.

Improved lower bounds for Hausdorff dimensions of projections in a specific dimension range.

Utilization of a variable coefficient local smoothing inequality in proofs.

Abstract

It is shown that if $A$ is a Borel subset of the first Heisenberg group, with Hausdorff dimension satisfying $2< \dim A < 3$, then the packing dimensions of vertical projections of $A$ are almost surely not less than $\dim A$, where both packing and Hausdorff dimensions are defined with respect to the Kor\'anyi metric. For the Hausdorff dimension of the projections, a weaker almost sure lower bound is obtained which improves the known bound in the range $2 < \dim A < \frac{1}{8}\left( 17 + \sqrt{33}\right) \approx 2.84$. The bound is slightly larger than $1+\frac{1}{2} \dim A$ and behaves similarly near $\dim A =2$. Both proofs rely on a variable coefficient local smoothing inequality.