Full-atom structure of Krzyż extremals

Determine whether every extremal function for the original Krzyż problem has exactly n atoms, and establish whether an extremal function with exactly n atoms exists for every n.

Background

The paper proves that every extremal function for the generalized coefficient problems is either a unimodular constant or a singular inner function with at most n atoms. It also observes that the supremum over candidate functions with fewer atoms is bounded by the supremum over candidate functions with n atoms.

For the original Krzyż problem, results of Lei and Zhang would imply the conjecture if every extremal function had exactly n atoms. However, the paper notes that neither the universal full-atom property nor the existence of an extremal function with exactly n atoms has been established.

References

Note, however, that it has not yet been established, even for the Kry.z problem, that every\/ extremal function must have the full number of atoms: $N=n$, nor that there exists an extremal function with this number of atoms.

— On extremal problems related to the Krzyż conjecture  (2610.23817 - Instanes et al., 16 Oct 2026) in Section 3, “Further properties of extremal functions”