slug: charlier-harju-puzynina-zamboni-question-two bibkey: charlier2015abelianbordered doi: 10.48550/arXiv.1501.07464 url: https://arxiv.org/abs/1501.07464v1 triage: theorem motivation_gids:
- D5/S1/Words/AbelianBorders/AbelianBorderQuestion.result
Weak Abelian Periodicity Does Not Force Finitely Many Weakly Abelian Unbordered Factors
Problem
Émilie Charlier, Tero Harju, Svetlana Puzynina and Luca Q. Zamboni, Abelian bordered factors and periodicity, arXiv:1501.07464v1, Section 7, Question 2: let w be an infinite bounded weakly abelian periodic word over a k-letter alphabet such that its graph G_w belongs to a k-dimensional cylinder with axis with rational coefficients, and each tangential line to G_w has points of G_w on it with bounded gaps. Does it follow that w has only finitely many weakly abelian unbordered factors?
Motivation
The theorem D5/S1/Words/AbelianBorders/AbelianBorderQuestion.result answers Question 2 negatively: an explicit word
over three letters satisfies all three hypotheses and has infinitely many weakly abelian unbordered factors.
Gap
Pre-registration issue 13129 records the literature screen: the paper’s Theorem 3 proves the forward implication and the binary equal-frequency case of the converse, the citing literature and the 2023 survey of abelian combinatorics on words do not settle Question 2, and the repository had no claim on it. This is a bounded negative finding.
Route
- With A = 110022222000111 and B = 012, the word w = B A^{e_1} B A^{e_2} … with unbounded exponents is a concatenation of blocks of length at most fifteen, all with letter frequencies (1/3, 1/3, 1/3), so it is bounded weakly abelian periodic.
- Projecting prefix Parikh vectors by T(p) = (p₀ − p₂, p₁ − p₂), the path of A traces a quadrilateral and the only new state of B lies strictly inside it; hence the graph lies in a cylinder around the line t(1, 1, 1), the tangential lines are the four axis-parallel lines through the exposed vertices, and each is visited in every window of eighteen indices.
- Each factor 12A^m0 starts and ends at the interior state, which is never an internal cut; the only chord through it joining two cut states forces equal prefix and suffix lengths, and the cut indices of its endpoints have the wrong residue modulo fifteen. Hence these factors are weakly abelian unbordered, and there are infinitely many of them.
Falsifier
The refutation would fail if some factor 12A^m0 had a nonempty proper prefix and a nonempty suffix with equal letter frequencies, or if some tangential line were visited only finitely often.
Evidence
The proof seat and an independent referee implementation checked K = 3, 5 and 7 on long prefixes, including all prefix and suffix pairs of the factors 12A^m0 for m ≤ 40, the cylinder radius and the tangential visit gaps.
Triage
theorem; the statement is Question 2 of arXiv:1501.07464v1, answered negatively.
- Refuted (formalized): Question 2 has a negative answer over a three-letter alphabet.
- Proved (paper): the same construction works for every odd K ≥ 3 with A = 1²0²2^{K+2}0^K1^K.
- Open: Question 1 of the same paper (whether abelian bordered long factors force abelian periodicity).
ASSUMED-UNVERIFIED
The literature screen is limited to the arXiv record of arXiv:1501.07464, its citing papers and the 2023 survey of abelian combinatorics on words, web and GitHub searches and the repository checks.