All projections of a typical Cantor set are Cantor sets

Abstract

In 1994, John Cobb asked: given $N>m>k>0$, does there exist a Cantor set in $\mathbb R^N$ such that each of its projections into $m$-planes is exactly $k$-dimensional? Such sets were described for $(N,m,k)=(2,1,1)$ by L.Antoine (1924) and for $(N,m,m)$ by K.Borsuk (1947). Examples were constructed for the cases $(3,2,1)$ by J.Cobb (1994), for $(N,m,m-1)$ and in a different way for $(N,N-1,N-2)$ by O.Frolkina (2010, 2019), for $(N,N-1,k)$ by S.Barov, J.J.Dijkstra and M.van der Meer (2012). We show that such sets are exceptional in the following sense. Let $\mathcal C(\mathbb R^N)$ be a set of all Cantor subsets of $\mathbb R^N$ endowed with the Hausdorff metric. It is known that $\mathcal C(\mathbb R^N)$ is a Baire space. We prove that there is a dense $G_δ$ subset $\mathcal P \subset \mathcal C(\mathbb R^N)$ such that for each $X\in \mathcal P$ and each non-zero linear subspace $L \subset \mathbb R^N$, the orthogonal projection of $X$ into $L$ is a Cantor set. This gives a partial answer to another question of J.Cobb stated in the same paper (1994).