These notes aim at illustrating some results achieved in the geometric measure theory in
Carnot groups. They are an extended version of a part of the course Geometric Measure …

Abstract Sixty years ago, John Marstrand published a paper which, among other things,
relates the Hausdorff dimension of a plane set to the dimensions of its orthogonal …

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We study the behavior of the Hausdorff dimension of sets in the Heisenberg group Hn, n∈
N, with a sub-Riemannian metric under projections onto horizontal and vertical subgroups …

The purpose of this paper is to introduce and study some basic concepts of quantitative
rectifiability in the first Heisenberg group $\Bbb {H} $. In particular, we aim to demonstrate …

A general approach to compute the spherical measure of submanifolds in homogeneous
groups is provided. We focus our attention on the homogeneous tangent space, that is a …

It is shown that vertical projections in the Heisenberg group of sets of dimension strictly
greater than 3 almost surely have positive area. The proof uses the point-plate incidence …

This is a survey on recent developments on the Hausdorff dimension of projections and
intersections for general subsets of Euclidean spaces, with an emphasis on estimates of the …

Let {π e: H→ W e: e∈ S 1} be the family of vertical projections in the first Heisenberg group
H. We prove that if K⊂ H is a Borel set with Hausdorff dimension dim H⁡ K∈[0, 2]∪{3}, then …

We explore the effect of vertical projections on the Hausdorff dimension of sets in the
Heisenberg group endowed with the Korányi distance. It is known that the dimension of an …