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

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. J\"{a}rvenp\"{a}\"{a}. An explicit construction follows, demonstrating that these distance sets bounds are sharp for a large class of polyhedral norms.