Cantor sets with high-dimensional projections

TL;DR

This paper explores the properties of Cantor sets in higher dimensions, demonstrating how their projections can be manipulated to be connected or of specific dimensions, and extends classical tameness results.

Contribution

It introduces new constructions of Cantor sets with prescribed projection properties and extends tameness criteria for zero-dimensional compacta in higher dimensions.

Findings

Antoine's necklace can have all projections connected and one-dimensional.

Any Cantor set in  can be isotoped so that projections are ()-dimensional.

Zero-dimensional compacta with certain projection properties are tame.

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.