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Determine the intersection {1/n!} ∩ C for the middle-third Cantor set

Determine the complete description of the set {1/n! : n ∈ N} ∩ C, where C denotes the middle-third Cantor set in [0,1]; in particular, ascertain whether this intersection equals {1}.

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Background

In discussing intersections of sequences with Cantor sets, the authors explicitly note that the structure of {1/n!} ∩ C is unknown. They revisit this again in their concluding questions, asking whether the intersection equals {1}.

The body of the paper develops arithmetic tools to analyze intersections with Cantor sets for various digit sets, but a definitive resolution for the factorial sequence intersected with the middle-third Cantor set is not provided.

References

For instance, we even do not know the structures of the following intersections: $$\left{\dfrac{1}{n2}:n\in \mathbb{N}\right}\cap C \quad\mbox{and}\quad \left{\dfrac{1}{n!}:n\in \mathbb{N}\right}\cap C.$$

— On the intersection of Cantor set with the unit circle and some sequences (2507.16510 - Jiang et al., 22 Jul 2025) in Introduction (Section 1)