Sticky Cantor sets

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $K\subseteq[0,1]$ be the middle-thirds Cantor set. An embedded Cantor set in $\mathbb R^n$ means a subset $C$ homeomorphic to $K$; it need not lie on a line. Say $C$ can be moved off itself at scale $\varepsilon>0$ if there is a homeomorphism $h:\mathbb R^n\to\mathbb R^n$ satisfying \[ \|h(x)-x\|<\varepsilon\quad\text{for every }x\in\mathbb R^n, \qquad h(C)\cap C=\varnothing. \] Call $C$ sticky in this small-homeomorphism sense if this fails at some scale $\varepsilon>0$.

Statement recorded in the supplied source.

For each $n\ge1$, is every embedded Cantor set movable off itself at every scale? Equivalently, determine exactly which dimensions admit a $C$ and an $\varepsilon>0$ such that every ambient homeomorphism moving each point by less than $\varepsilon$ has $h(C)\cap C\ne\varnothing$. [S2]

Short English statement

Determine in which Euclidean dimensions an embedded Cantor set cannot be separated from itself by a sufficiently small ambient homeomorphism.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3422 (OPG-37293). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Sticky Cantor sets, OPG-37293. Original problem statement, full mathematical discussion, bibliography and comments. https://www.openproblemgarden.org/op/sticky_cantor_sets.