On the packing dimension of exceptional sets of projections

Ryan E. G. Bushling

Published 2022-11-03Version 1

Let $A \subseteq \mathbb{R}^n$ be analytic. An exceptional set of projections for $A$ is a set of $k$-dimensional subspaces of $\mathbb{R}^n$ onto which the orthogonal projection of $A$ has "unexpectedly low" Hausdorff dimension. The famous projection theorems of Mattila (1975) and Falconer (1982) place upper bounds on the Hausdorff dimensions of exceptional sets, considered as subsets of the Grassmannian $\mathbf{Gr}(n,k)$. A 2015 result of Orponen bounds the packing dimension of the exceptional set in the case that $n = 2$, $k = 1$, and $A$ is self-similar or homogeneous. Our purpose is to extend Orponen's result to the case of arbitrary $0 < k < n$.