If $A\subset\mathbb{R}^2$ is a Borel measurable set and $p\_\theta$ is projection onto the line spanned by $(\cos\theta,\sin\theta)$, then it is well known that for almost every $\theta\in[0,2\pi]$, $p\_\theta(A)$ has Hausdroff dimension the min of 1 and the Hausdorff dimension of $A$.


My question is: let's say $A$ has Hausdorff dimension 1, what would be an example where $p\_\theta(A)$ has Hausdorff dimension <1 for at least 3 different $\theta\in[0,2\pi]$? What if the number 3 is replaced by countable infinity?



Exceptional set for Marstrand's projection theorem
Theorem 1.1 of Orponen's paper "On the packing dimension and category of exceptional sets of orthogonal projections" contains a (relatively short) construction of a compact set $K\subseteq\mathbb{R}^2$ with $\infty>\mathcal{H}^1(K)>0$ and a dense $G\_\delta$ set $\Omega\subseteq[0,2\pi)$ such that $p\_\theta(K)$ has zero Hausdorff dimension, for all $\theta\in\Omega$. I would be interested if an expert here had an easy example for the simpler version of the question that we're asking here, however.


