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 ${\mathbb{R}}^n$, $n⩾3$, 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 $X\subset {\mathbb{R}}^n$, $n⩾2$, 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.

Cited by in corpus (3)

• All projections of a typical Cantor set are Cantor sets
• A new simple family of Cantor sets in ${\mathbb{R}}^3$ all of whose projections are one-dimensional
• Most Cantor sets in ${\mathbb{R}}^N$ are in general position with respect to all projections