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Packing dimension of vertical projections in the Heisenberg group

Terence L. J. Harris

TL;DR

The paper proves that vertical projections in the first Heisenberg group preserve packing dimension for Borel sets with Kor\'anyi dimension strictly between 2 and 3, in the sense that almost every vertical projection has packing dimension at least the original dimension. The approach combines a quantitative projection theorem (domain via Kor\'anyi metric, codomain via Euclidean metric), an $L^p$ inequality for projections obtained via a variable-coefficient local smoothing inequality, and a slicing/intersection framework to transfer projection bounds into lower bounds on fiber intersections. This yields two key results: a packing-dimension version of a projection theorem and an intersection theorem that together imply the main packing-dimension lower bound for vertical projections. The work advances understanding of dimension theory in sub-Riemannian spaces and illustrates how harmonic-analytic tools (local smoothing, Fourier-integral operators, and projection/intersection analysis) can be adapted to the Heisenberg setting.

Abstract

It is shown that if $A$ is a Borel subset of the first Heisenberg group, with Hausdorff dimension satisfying $2< \dim A < 3$, then the packing dimensions of vertical projections of $A$ are almost surely not less than $\dim A$, where both packing and Hausdorff dimensions are defined with respect to the Korányi metric. The proof relies on a variable coefficient local smoothing inequality.

Packing dimension of vertical projections in the Heisenberg group

TL;DR

The paper proves that vertical projections in the first Heisenberg group preserve packing dimension for Borel sets with Kor\'anyi dimension strictly between 2 and 3, in the sense that almost every vertical projection has packing dimension at least the original dimension. The approach combines a quantitative projection theorem (domain via Kor\'anyi metric, codomain via Euclidean metric), an inequality for projections obtained via a variable-coefficient local smoothing inequality, and a slicing/intersection framework to transfer projection bounds into lower bounds on fiber intersections. This yields two key results: a packing-dimension version of a projection theorem and an intersection theorem that together imply the main packing-dimension lower bound for vertical projections. The work advances understanding of dimension theory in sub-Riemannian spaces and illustrates how harmonic-analytic tools (local smoothing, Fourier-integral operators, and projection/intersection analysis) can be adapted to the Heisenberg setting.

Abstract

It is shown that if is a Borel subset of the first Heisenberg group, with Hausdorff dimension satisfying , then the packing dimensions of vertical projections of are almost surely not less than , where both packing and Hausdorff dimensions are defined with respect to the Korányi metric. The proof relies on a variable coefficient local smoothing inequality.
Paper Structure (5 sections, 7 theorems, 134 equations)

This paper contains 5 sections, 7 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Motivation for the proof of Theorem \ref{['etccorollary1']} and Theorem \ref{['etccorollary']}
  3. A quantitative projection theorem with Korányi metric in domain and Euclidean metric in co-domain
  4. An Lp inequality for vertical projections in the Euclidean metric
  5. An intersection theorem

Key Result

Theorem 1.1

Let $A \subseteq \mathbb{H}$ be a Borel set with $2 < \dim A < 3$. Then $\dim_P P_{\mathbb{V}_{\theta}^{\perp}}(A) \geq \dim A$ for a.e. $\theta \in [0, \pi)$.

Theorems & Definitions (14)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • Remark
  • proof
  • Lemma 4.1
  • proof
  • ...and 4 more