Minimal reversible exclusion sets for Eisenstein integers

Classify the minimal reversible exclusion sets of length two over the Eisenstein integers.

Background

The paper develops exclusion-set criteria for integer and Gaussian-integer continued fractions using continued-fraction dynamics and the corresponding Farey graphs. It observes that the analogous Farey graph for Eisenstein integers is the one-skeleton of a tessellation of hyperbolic three-space by tetrahedra.

The authors identify the Eisenstein-integer case as a separate unresolved problem and suggest that its geometric structure could make the computation of exclusion sets more tractable than in the Gaussian-integer case.

References

Another, separate open problem is to determine exclusion sets for Eisenstein integers. \begin{question} Classify the minimal reversible exclusion sets over the Eisenstein integers of length two. \end{question}

— Optimal local convergence criteria for integer and Gaussian integer continued fractions  (2608.13199 - Short et al., 13 Aug 2026) in Section 4, immediately after the question on Gaussian-integer exclusion sets