The effect of projections on dimension in the Heisenberg group
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.
scientific article; zbMATH DE number 6176457
scientific article