📝 SummarXiv

Abstract

It is shown that if $A$ is a Borel subset of the first Heisenberg group contained in either a vertical subgroup or a horizontal plane, then vertical projections almost surely do not decrease the Hausdorff dimension of $A$, with respect to the Kor\'anyi metric. For general Borel sets, it is shown that if $\dim A >2 $, then vertical projections of $A$ almost surely have dimension at least $(4+\dim A)/3$. This improves the known bound in the general case when $2 < \dim A < 13/5$. The horizontal case and the general case both rely on a variable coefficient local smoothing inequality.