Exceptional set for Marstrand's projection theorem

Question

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?

Answer 1

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.

Answer 2

I'm aware of one canonical example with dense, countably infinite exceptional set, which is the "one-dimensional Sierpinski triangle" studied by Furstenberg, Kenyon and Hochman.

Define three transformations $T_1, T_2, T_3 \colon \mathbb{R}^2 \to \mathbb{R}^2$ by $$T_1\begin{pmatrix} x\\y\end{pmatrix} :=\frac{1}{3}\begin{pmatrix} x\\y\end{pmatrix},\quad T_2\begin{pmatrix} x\\y\end{pmatrix}:=\frac{1}{3}\begin{pmatrix} x\\y\end{pmatrix} + \begin{pmatrix}1\\0\end{pmatrix}, \quad T_3\begin{pmatrix} x\\y\end{pmatrix}=\frac{1}{3}\begin{pmatrix} x\\y\end{pmatrix}+\begin{pmatrix}0\\1\end{pmatrix}.$$ By Hutchinson's theorems on self-similar sets, there exists a unique nonempty compact set $X\subset \mathbb{R}^2$ satisfying $X=\bigcup_{i=1}^3 T_iX$. This set looks somewhat like (the right-angled version of) the ordinary Sierpinski triangle, but it is totally disconnected and has much lower dimension: it has Hausdorff dimension 1, and it has positive and finite 1-dimensional Lebesgue measure. In unpublished work in the 1970s, Furstenberg showed that every orthogonal projection onto a line with rational slope $p/q$ satisfying $3\nmid p+q$ is an exceptional projection. The projection onto the vertical axis is also exceptional. Furstenberg's work was later described and extended by Richard Kenyon in a 1997 paper in Israel J. Math, where it was conjectured that those projections are the only exceptional projections. That conjecture was eventually proved by Michael Hochman in his 2014 Annals paper.

To show that those projections are exceptional is the easier part. Given such a projection $P$, one shows that the image $PX$ set satisfies a self-similarity equation like $$PX=\bigcup_{(i_1,\ldots,i_n) \in \{1,2,3\}^n} PT_{i_1}\circ \cdots \circ T_{i_n} PX$$ but where the union is over a proper subset of the set of all $3^n$ possible words of length $n$ -- say, using only $m<3^n$ words. This implies via Hutchinson's theorems that there is an upper bound $\log m / \log (3^n)<1$ for the Hausdorff dimension of $PX$. One does this by showing that there exist two finite sequences $(i_1,\ldots,i_n), (j_1,\ldots,j_n) \in \{1,2,3\}^n$ such that the two transformations $PT_{i_1}\circ T_{i_2}\circ\cdots\circ T_{i_n}P$ and $PT_{j_1}\circ T_{j_2}\circ\cdots\circ T_{j_n}P$ are identical. (Note the helpful identity $PT_i=PT_iP$ for every $i$.) This part of the argument can be found in Kenyon's paper and I think just boils down to a pigeonhole argument, where $3^n$ objects are arranged with distinct pairs having to be more than $1/3^n$ distance apart, or something like that. The result that all other projections are non-exceptional is much deeper and relies on later techniques introduced by Michael Hochman.