Infinitely many primitively complete integers

Determine whether, under the hypotheses of Theorem 2.9, there exist infinitely many primitively complete integers with respect to $(b,\mathcal C)$.

Background

The paper introduces complete and incomplete integers relative to a base b and a digit representation. It proves that, under the relevant hypotheses, infinitely many primitively incomplete integers exist.

The complementary abundance of primitively complete integers remains unresolved. This is the arithmetic formulation corresponding to the complementary spectral-eigenvalue question for canonical spectral pairs.

References

Under the above conditions in Theorem \ref{theonotd}, it is not clear whether there exist infinitely many primitively complete integers w.r.t. $(b, )$.

— Spectral eigenvalue problem of Cantor measures and Artin's primitive root conjecture  (2609.29038 - He et al., 24 Sep 2026) in Remark following Theorem 2.9, Section 2