Cantor-like set in $\mathbb{R}^2$ is purely 1-unrectifiable

Question

I was just reading about rectifiability, and an example of purely 1-unrectifiable set can be given as follows. Consider the Cantor-like set $C_{1/2}$ obtained like this:

$$[0,1] \to [0, 1/4] \cup [3/4,1] \to [0,1/16] \cup [ 3/16, 1/4] \cup [3/4 , 13/16] \cup [15/16, 1] \to \dots = C_{1/2} .$$

Then, it can be proved that this set has Hausdorff dimension of 1/2. And that $C=C_{1/2} \times C_{1/2}$ has Hausdorff dimension of 1. Now, I would like to prove that $C$ is purely 1-unrectifiable by using the characterization by projections on hyperplanes. The theorem says that

$$\text{C is } \mathcal{H}^k - \text{rectifiable} \iff \mathcal{H}^k (\pi (C))>0 \text{ for } \Theta \text{ almost all } \pi \in G(k,n), $$

where $\Theta$ is a Haar measure on the Grassmanians $G(k,n)$. Now, in this case we simply have $G(1,2)$, i.e. one dimensional vector spaces in the plane. My doubts are: how is $\Theta$ defined in this case? And is there a simple way to see that $\mathcal{H}^1 (\pi (C))=0$ for a set of lines in $G(1,2)$ of positive measure? Because when $\pi$ equals one of the two axis it is trivial, but this doesn't give me anything. I guess it's true that for every $\pi \in G(1,2)$ it holds $\mathcal{H}^1 (\pi (C))=0$, I was just wondering if there was an easy way to see this besides actually computing the projection.

Answer 1

For sets $K$ in the plane of finite one dimensional measure, there is a beautiful theorem by Besicovich:

$K$ is purely nonrectifiable iff there are two distinct lines so that the orthogonal projection of $K$ on these lines has zero length.

You can learn about this in [1], and it immediately implies that the four corner set $K=C_{1/2}\times C_{1/2}$ is purely nonrectifiable.

A direct elementary proof for this set is in [2], and that proof is reproduced in [3].

It is still unknown at what rate does the Favard length (average length of projections) of $\epsilon$-neighborhoods of $K$ tend to zero. The best bounds known are in [4] and [5].

[1] Falconer, K.J., 1986. The geometry of fractal sets (No. 85). Cambridge university press.

[2] Peres, Yuval, Károly Simon, and Boris Solomyak. "Fractals with positive length and zero Buffon needle probability." The American mathematical monthly 110, no. 4 (2003): 314-325.

[3] https://www.yuval-peres-books.com/fractals-in-probability-and-analysis/

[4] Nazarov, F., Peres, Y. and Volberg, A., 2011. The power law for the Buffon needle probability of the four-corner Cantor set. St. Petersburg Mathematical Journal, 22(1), pp.61-72.

[5] Bateman, M. and Volberg, A., 2010. AN ESTIMATE FROM BELOW FOR THE BUFFON NEEDLE PROBABILITY OF THE FOUR-CORNER CANTOR SET. Mathematical research letters, 17(5), pp.959-967.