Higher-Dimensional Analogues of Two Theorems of Orponen on Exceptional Sets

Abstract

Let $A \subseteq \mathbb{R}^n$ be a Borel set of Hausdorff dimension $\dim_{\mathrm H} A = s$ and, for each $m$-plane $V$ in the Grassmannian $\mathbf{Gr}(n,m)$, let $\pi_V : \mathbb{R}^n \to V$ be the orthogonal projection of $\mathbb{R}^n$ onto $V$. \textit{Marstrand's projection theorem} states that $\dim_{\mathrm H} \pi_V(A) = \min \, \{s,m\}$ for almost every $V \in \mathbf{Gr}(n,m)$, and \textit{Marstrand's slicing theorem} states that the set of $\mathbf{x} \in V$ such that $\dim_{\mathrm H} \big( \pi_V^{-1}(\mathbf{x}) \cap A \big) = s-m$ has positive Lebesgue measure for almost every $V \in \mathbf{Gr}(n,m)$ provided $s > m$. Informally, for all but a few subspaces $V$, the shadow of $A$ on $V$ has the largest possible dimension, and there are many sections of $A$ orthogonal to $V$ of essentially maximal dimension. In this dissertation, we prove two results concerning the sizes of \textit{exceptional sets} of projections and slices—the null sets of parameters $V$ for which the conclusion of Marstrand's projection or slicing theorem fails, respectively. These are, in particular, higher-dimensional analogues of two theorems of Tuomas Orponen. The first result states that, if linear maps are \textit{almost dimension conserving} for $A$, then the exceptional sets of orthogonal projections of $A$ have small packing dimension. This applies, for example, to (weakly) Furstenberg homogeneous sets and to certain self-similar and graph-directed sets. The second concerns slices by fibers of the \textit{generalized projections} of Peres and Schlag. Roughly, the conclusion of Marstrand's slicing theorem holds for very general families of nonlinear projections, and the exceptional sets of slices have small Hausdorff dimension. In fact, not only is the set of exceptional slices of $A$ small, but the union of the exceptional sets of all positive-$\mathcal{H}^s$-measure Borel \textit{subsets} of $A$ is small.