On factorial reciprocals in Cantor sets

TL;DR

This paper investigates the intersection of factorial reciprocals with Cantor sets, proving only finitely many such reciprocals lie in the set and providing a method to determine them.

Contribution

It generalizes the result to missing-digit sets, showing finitely many factorial reciprocals are contained and can be explicitly identified.

Findings

Only 1 and 1/5! are in the middle-third Cantor set.

The method applies to general missing-digit sets.

Finitely many factorial reciprocals are contained in these 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.