The effect of projections on dimension in the Heisenberg group
type: Item
instance of: scholarly article
title: The effect of projections on dimension in the Heisenberg group
zbMATH Open document ID: 1279.28007
DOI: 10.4171/RMI/725
author: Zoltán M. Balogh
author: Estibalitz Durand-Cartagena
author: Jeremy T. Tyson
author: Katrin Fässler
author: Pertti Mattila
published in: Revista Matemática Iberoamericana
publication date: 2013-06-17T00:00:00Z
review text: Let \({\mathbb H}\) be the Heisenberg group identified with \({\mathbb C}\times{\mathbb R}\). Its points are denoted by \((z,t)\), the group law is defined by the equality \((z,t)*(\zeta,\tau)=(z+\zeta, t+\tau +2\Im(z\overline{\zeta}))\), and the metric is \(d(p,q)=\|q^{-1}*p\|_{\mathbb H}\), where \(\|(z,t)\|_{\mathbb H} = (|z|^4 +t^2)^{1/4}\). For a fixed \(\theta\in[0,\pi)\), let \({\mathbb V}_{\theta}\) be the one-dimensional subspace of \({\mathbb R}^3\) spanned by the vector \((e^{i\theta}, 0)\), and let \({\mathbb W}_{\theta}\) be its Euclidean orthogonal complement. The mappings \(p_{{\mathbb V}_{\theta}}: {\mathbb H}\mapsto {\mathbb V}_{\theta}\) and \(p_{{\mathbb W}_{\theta}}: {\mathbb H}\mapsto {\mathbb W}_{\theta}\) are horizontal and vertical projections. The authors obtain universal (i.e., valid for all sets and angles) and almost sure (i.e., valid for all sets and for almost all angles) estimates for the Hausdorff dimensions of horizontal and vertical projections of a Borel set \(A\subset{\mathbb H}\). For instance, they prove that \[ \text{dim}p_{{\mathbb V}_{\theta}}(A)\geq\max\{0, \text{dim}_{\mathbb H}A-3\} \] for all \(\theta\) (universal lower bound), and \[ \text{dim}p_{{\mathbb V}_{\theta}}(A)\geq\max\{0, \min\{\text{dim}_{\mathbb H}A-2, 1\}\} \] for almost all \(\theta\) (almost sure lower bound). Here \(\text{dim}\) and \(\text{dim}_{\mathbb H}\) stand for the Hausdorff dimensions with respect to the Euclidean and the Heisenberg metric, respectively.
Mathematics Subject Classification ID: 28A78
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 43A80
zbMATH DE Number: 6176457
zbMATH Keywords: Heisenberg group
zbMATH Keywords: projection theorems
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: energy integrals
reviewed by: Boris A. Kats
MaRDI profile type: Publication
Wikidata QID: Q110007345
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description: scientific article; zbMATH DE number 6176457