On the packing dimension of distance sets with respect to $C^1$ and polyhedral norms

TL;DR

The paper establishes packing-dimension lower bounds for distance sets relative to polyhedral and $C^1$ norms in $\mathbb{R}^d$: for any set $E$ and any such norm, there exists a point $x\in E$ with $\dim_{\mathrm{P}}\Delta_x^*(E) \ge \frac{1}{d}\dim_{\mathrm{P}} E - \varepsilon$, implying $\dim_{\mathrm{P}}\Delta^*(E) \ge \frac{1}{d}\dim_{\mathrm{P}} E$. The core method combines a nonlinear projection estimate that extends Järvenpää’s orthogonal projection results with a weak transversality framework tailored to both polyhedral and $C^1$ norms. For polyhedral norms with rationally dependent face normals, the authors construct explicit examples showing sharpness of the bound, achieving $\dim_{\mathrm{P}}\Delta^P(E) = \frac{1}{d}\dim_{\mathrm{P}} E}$. Together, these results extend Falconer-type distance-set bounds to the packing-dimension setting under broad norm classes and provide a versatile toolkit via nonlinear projections and transversality arguments.

Abstract

We prove that, for every polyhedral or $C^1$ norm on $\mathbb{R}^d$ and every set $E \subseteq \mathbb{R}^d$ of packing dimension $s$, the packing dimension of the distance set of $E$ with respect to that norm is at least $\tfrac{s}{d}$. One of the main tools is a nonlinear projection theorem extending a result of M. Järvenpää. An explicit construction follows, demonstrating that these distance sets bounds are sharp for a large class of polyhedral norms.

Theorems & Definitions (29)

• Theorem 1: altaf2023distance Theorems 1.1 & 1.4
• Theorem 1.1
• Theorem 1.2
• Definition 1.1: Weak transversality
• Proposition 1.3
• Theorem 3.1: Järvenpää jarvenpaa1994upper Theorem 3.4 & Corollary 3.5
• proof : Proof of Proposition \ref{['prop:nonlinear-jarvenpaa']}
• Lemma 4.1
• proof
• Proposition 4.2