Weak Abelian Periodicity Does Not Force Finitely Many Unbordered Factors
Abstract
An explicit ternary word gives a negative answer to Question 2 of Charlier, Harju, Puzynina, and Zamboni.
Theorem 1.1 (Question 2 has a negative answer).
Proof. Machine-checked in Lean as D5/S1/Words/AbelianBorders/AbelianBorderQuestion.result (✓ std3). ∎
Resolves. Problems/charlier-harju-puzynina-zamboni-question-two (refuted) by D5/S1/Words/AbelianBorders/AbelianBorderQuestion.result.
Source. Repository-derived.
Acknowledgement. Émilie Charlier, Tero Harju, Svetlana Puzynina, Luca Q. Zamboni (2015). Abelian bordered factors and periodicity. DOI: 10.48550/arXiv.1501.07464. URL: https://arxiv.org/abs/1501.07464v1.
Commentary.
The ternary word has a bounded decomposition into equal-frequency blocks and satisfies the cylinder and tangential-line conditions, yet it has infinitely many nonempty weakly abelian unbordered factors. This answers Question 2 negatively even when the suffix of a border may be the entire word.