Rationality criterion for non-integer base representations

Determine an analogue, for non-integer bases, of the criterion stating that a number is rational if and only if its positional representation is eventually periodic.

Background

For integer bases with the standard alphabet, rational numbers are characterized by eventual periodicity of their classical positional representations. The paper notes that the corresponding characterization has not been established for non-integer bases, leaving unresolved whether an analogous criterion exists and what form it should take.

References

For non-integer bases, however, no analogous criterion is known.

— Some applications of binary numeration systems with nonzero redundancy to the theory of locally complex functions  (2609.34746 - Vaskevych et al., 28 Sep 2026) in Section 1, Introduction