Simultaneous Sturmian representations in two bases

Determine whether only finitely many integers have both a Sturmian base-b representation and a Sturmian ternary representation.

Background

The paper studies restrictions on the base-b representations of integers whose digit words are Sturmian, proving strong arithmetic constraints for such integers and, in particular, results concerning their prime divisors. The concluding question asks whether the simultaneous occurrence of Sturmian representations in base b and in base 3 can happen for only finitely many integers. The question concerns the intersection of two structurally constrained digital classes and is not resolved in the paper.