On $\frac{1}{n!}$ in Cantor sets

Abstract

Let $C$ be the middle-third Cantor set. We show that \[\left\{\frac{1}{n!}: n\in\mathbb{N}\right\}\cap C=\left\{1, \frac{1}{5!}\right\}.\] This answers a question recently posed by Jiang [J. Lond. Math. Soc., 2026, published online]. Our approach generalizes to general missing-digit sets, showing that, in any such set, there are only finitely many elements of the form $\frac{1}{n!}$, all of which can be effectively determined.

Theorems & Definitions (10)

• Theorem 1.2
• Definition 1.3
• Theorem 1.4
• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• Lemma 2.4
• proof : Proof of Lemma \ref{['lemordMn']}
• proof : Proof of Theorem \ref{['thmCn!']}