On the packing dimension of unions and extensions of $k$-planes

TL;DR

This paper investigates the packing dimension of unions and extensions of subsets of $k$-planes in Euclidean space, introducing effective dimension concepts and improving bounds especially for hyperplanes.

Contribution

It introduces a notion of effective dimension on Grassmannians and extends packing dimension results to unions and extensions of $k$-planes, including the case of hyperplanes.

Findings

Introduces effective dimension on Grassmannian and affine Grassmannian.

Establishes bounds on packing dimension for unions of $k$-planes.

Improves bounds for the case when $k=n-1$.

Abstract

We study the packing dimension of unions of subsets of $k$-planes in $\mathbb{R}^n$ using tools from algorithmic information theory, obtaining an analog of a result of H\'era and a mild generalization of a recent result of Fraser. Along the way, we introduce a notion of effective dimension on the Grassmannian and affine Grassmannian, and we establish several useful algorithmic and geometric tools in this setting. Additionally, we consider how the packing dimension of the union of certain subsets of $k$-planes changes when the subsets are extended to the entire $k$-plane. Finally, we improve the above bounds for unions and extensions in the special case that $k=n-1$.