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Statement

A Cantor set is a compact, totally disconnected, perfect subset of $\mathbb R$ (p. 9). A set $X\subseteq\mathbb R$ is topologically universal if for every dense $G_\delta$ subset $G$ of $\mathbb R$ there are $\lambda\in\mathbb R\setminus\{0\}$ and $t\in\mathbb R$ with $\lambda X+t\subset G$ (p. 14).

Theorem 4.2 (Jung--Lai; p. 14). If $K$ is a Cantor set in $\mathbb R$, then there are a Cantor set $\widetilde K$ in $\mathbb R$ and $\delta>0$ such that

$$K\cap(\lambda\widetilde K+t)\ne\varnothing \quad\text{for all }\lambda\in\Bigl(\frac1{1+\delta},1+\delta\Bigr) \text{ and all }t\in(-\delta,\delta).$$

In particular, $K$ is not topologically universal.

The survey records that Gallagher, Lai and Weber first proved that no Cantor set in $\mathbb R^d$ is topologically universal, and presents this theorem as another proof on the line (p. 14). It notes that the sets $\widetilde K$ so produced have positive Lebesgue measure, so the approach gives nothing on measure universality (p. 15). Topologically universal sets are exactly the sets of strong measure zero, by a result of Jung and Lai that the survey cites (p. 15).

Source. Yeonwook Jung, Chun-Kit Lai and Yuveshen Mooroogen, Fifty years of the Erdős similarity conjecture, arXiv:2412.11062v2 (1 January 2025), whose labels and page numbers are cited here; the edition is identified on the source card.

Read depth. Claims checked: the statement and definitions were read clause by clause on pp. 9 and 14, and the survey's proof (pp. 14--15) was read; nothing here is independently reviewed.

Proof pointer

Pages 14--15. The containment lemma (Lemma 4.1, p. 14) says that two Cantor sets meet when the convex hull of the first lies in that of the second and, at every level $n$ of their binary constructions, every level-$n$ gap of the second is shorter than every level-$n$ gap of the first. Choose $\widetilde K$ whose hull strictly contains that of $K$ and whose level-$n$ gaps are below half the shortest level-$n$ gap of $K$; for $\delta$ small the same holds with $\lambda K+t$ in place of $K$, and the lemma gives the intersection. For the second claim, $M=\bigcup_{(a,b)\in\mathbb Q^2} (a\widetilde K+b)$ meets every $\lambda K+t$; $M$ is a countable union of nowhere dense closed sets, so its complement is a dense $G_\delta$ set that contains no nontrivial affine copy of $K$.

Dependencies

Lemma 4.1 (the containment lemma, p. 14), from Y. Jung and C.-K. Lai, Interior of certain sums and continuous images of very thin Cantor sets (2024), and the Baire category theorem.

Bears on

• Problem 120: the topological analogue only, with dense $G_\delta$ sets in place of sets of positive measure. A dense $G_\delta$ set can be null, and the survey notes that the construction gives no measure statement (p. 15), so the theorem says nothing about Problem 120 directly.