arXiv subjects

Fernando Roman-Garcia

Publications and source records attributed to Fernando Roman-Garcia.

3 recordsLinked to original sources

arXiv2020

Dimension Distortion by Right Coset Projections in the Heisenberg Group

We study the family of vertical projections whose fibers are right cosets of horizontal planes in the Heisenberg group, $\mathbb{H}^n$. We prove lower bounds for Hausdorff dimension distortion of sets under these mappings, with respect to the Euclidean metric and also the natural quotient metric which we show behaves like the Euclidean metric in this context. Our bounds are sharp in a large part of the dimension range, and we give conjectural sharp lower bounds for the remaining range. Our approach also lets us improve the known almost sure lower bound for the standard family of vertical projections in $\mathbb{H}^n$ for $n \geq 2$.

math.MG↗

arXiv2020

A Fourier Coefficients Approach to Hausdorff Dimension in the Heisenberg Group

This paper establishes connections between the group-Fourier transform and the geometry of measures in the Heisenberg group. Firstly, it is shown that if the Fourier transform of a compactly supported, finite, Radon measure is square integrable, then the measure must have a square integrable density. If it's Fourier transform is integrable, the the measure must have a continuous density. In addition, an alternative formulation of the Fourier transform on the Heisenberg group is used to show that energies of measures can be computed via integrals on an appropriate frequency space. This in turns opens the possibility of using Fourier methods in the computation of Hausdorff dimension of sets.

math.FA↗

arXiv2020

Intersection of projections and slicing theorems for the isotropic Grassmannian and the Heisenberg group

This paper studies the Hausdorff dimension of the intersection of isotropic projections of subsets of $\mathbb{R}^{2n}$, as well as dimension of intersections of sets with isotropic planes. It is shown that if $A$ and $B$ are Borel subsets of $\mathbb{R}^{2n}$ of dimension greater than m, then for a positive measure set of isotropic m-planes, the intersection of the images of $A$ and $B$ under orthogonal projections onto these planes have positive Hausdorff $m$-measure. In addition, if $A$ is a measurable set of Hausdorff dimension greater than $m$, then there is a set $B\subset\mathbb{R}^{2n}$ with $\dim B\leq m$ such that for all $x\in\mathbb{R}^{2n}\setminus B$ there is a positive measure set of isotropic m-planes for which the translate by $x$ of the orthogonal complement of each such plane, intersects $A$ on a set of dimension $\dim A-m$. These results are then applied to obtain analogous results on the $n^{th}$ Heisenberg group.

math.MG↗