Classification of ordered Gaussian-integer exclusion sets

Classify the minimal ordered exclusion sets of length two over the Gaussian integers, extending the paper’s classification of minimal reversible exclusion sets.

Background

The paper proves that there are exactly two minimal reversible exclusion sets of length two over the Gaussian integers. Reversibility identifies a pair with its reversal, whereas an ordered exclusion set need not have this symmetry.

The authors explicitly state that the classification problem without the reversibility requirement remains unresolved. They note that a first step would be to find a length-two exclusion set that excludes fewer continued fractions than the previously known set of Dani and Nogueira.

References

Theorem~\ref{theoremD} classifies the minimal reversible exclusion sets over the Gaussian integers of length two; the problem of classifying the ordered exclusion sets remains open. \begin{question} Classify the minimal exclusion sets over the Gaussian 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 proof of Theorem D