It is shown that if $A$ is a Borel subset of the first Heisenberg group contained in some vertical subgroup, then vertical projections almost surely do not decrease the Hausdorff dimension of $A$.
On the distance sets spanned by sets of dimension $d/2$ in $\mathbb{R}^d$
On the Hausdorff dimension of Furstenberg sets and orthogonal projections in the plane
On the Dimension of Kakeya Sets in the First Heisenberg Group
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