Cantor sets with high-dimensional projections

Abstract

In 1994, J.Cobb constructed a tame Cantor set in $\mathbb R^3$ each of whose projections into $2$-planes is one-dimensional. We show that an Antoine's necklace can serve as an example of a Cantor set all of whose projections are one-dimensional and connected. We prove that each Cantor set in $\mathbb R^n$, $n\geqslant 3$, can be moved by a small ambient isotopy so that the projection of the resulting Cantor set into each $(n-1)$-plane is $(n-2)$-dimensional. We show that if $X\subset \mathbb R^n$, $n\geqslant 2$, is a zero-dimensional compactum whose projection into some plane $\Pi\subset \mathbb R^n$ with $\dim \Pi \in \{1, 2, n-2, n-1\}$ is zero-dimensional, then $X$ is tame; this extends some particular cases of the results of D.R.McMillan, Jr. (1964) and D.G.Wright, J.J.Walsh (1982). We use the technique of defining sequences which comes back to Louis Antoine.