Packing dimension of vertical projections in the Heisenberg group

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 Korányi 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.

34 pages. V8: Proof fixed by using packing instead of Hausdorff dimension. V10: Added partial result for Hausdorff dimension which restores result from V4 with small improvement