Bounding the dimension of exceptional sets for orthogonal projections

TL;DR

This paper investigates the size of the set of projections of a set in Euclidean space where the projected dimension drops below a certain threshold, improving bounds and providing sharp results in specific cases.

Contribution

It improves existing estimates on the dimension of exceptional projection sets and fully resolves the problem for three-dimensional space.

Findings

Improved bounds on the dimension of exceptional sets for projections.

Sharp estimates established for the cases when k=1 and k=n-1.

Complete resolution of the problem in three-dimensional space.

Abstract

It is well known that if $A \subseteq \mathbb{R}^n$ is an analytic set of Hausdorff dimension $a$, then $\dim_H(\pi_VA)=\min\{a,k\}$ for a.e.\ $V\in G(n,k)$, where $G(n,k)$ denotes the set of all $k$-dimensional subspaces of $\mathbb{R}^n$ and $\pi_V$ is the orthogonal projection of $A$ onto $V$. In this paper we study how large the exceptional set \begin{equation*} \{V\in G(n,k) \mid \dim_H(\pi_V A) < s\} \end{equation*} can be for a given $s\le\min\{a,k\}.$ We improve previously known estimates on the dimension of the exceptional set, and we show that our estimates are sharp for $k=1$ and for $k=n-1$. Hence we completely resolve this question for $n=3$.