Lebesgue measure of the projections of a compact set

Question

Does the following statement follow easily from any known theorem?

Statement: There exists a compact set $K$ in $\mathbb{R}^2$ such that its projection to the $x$-axis is the $[0,1]$ interval, but in all other directions its projections are (Lebesgue) nullsets.

Context: I think I created a proof for this and I wonder whether I found something new.

Answer 1

I found the answer. There exists such a set. For direct proof, see https://www.sciencedirect.com/science/article/pii/S0723086908000066 (Appendix A)

Moreover, Michel Talagrand proved the following generalization of this:

Let $K$ be a compact set. Let $f_K: S^1\to \mathbb{R}$, $f_K(\alpha)=\lambda(proj_\alpha(K))$ (the Lebesgue-measure of the orthogonal projection in the direction $\alpha$). Then the functions which can be obtained in the form $f_K$ for some compact set $K$ are exactly the nonnegative upper semicontinuous functions.

Reference: Sur la mesure de la projection d'un compact et certaines familles de cercles" Bull. Sci. Math. (2) 104 (1980), no. 3, 225–231.

I wasn't able to read the article yet. However many experts cite it so it should be ok.