Bounding the dimension of exceptional sets for orthogonal projections

It is well known that if $A\subseteq {\mathbb{R}}^n$ is an analytic set of Hausdorff dimension $a$, then ${\dim }_H({\pi }_{V}A)=\min \{a,k\}$ 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(π_V A) < s} \end{equation*} can be for a given $s\le \min \{a,k\}\mathrm{.}$ 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 $n=3$.