bibkey: charlier2015abelianbordered authors: Émilie Charlier, Tero Harju, Svetlana Puzynina, Luca Q. Zamboni year: 2015 title: “Abelian bordered factors and periodicity” doi: 10.48550/arXiv.1501.07464 url: https://arxiv.org/abs/1501.07464v1 claim: “Infinite words whose long factors are abelian or weakly abelian bordered; Theorem 3 shows that finitely many weakly abelian unbordered factors force bounded weak abelian periodicity with geometric conditions on the graph, and Section 7, Question 2 asks for the converse.” strata_touched:

• D5/S1/Words/AbelianBorders/AbelianBorderQuestion license: citation-only triage: anchor

Charlier, Harju, Puzynina and Zamboni, abelian bordered factors and periodicity

The paper studies infinite words all of whose sufficiently long factors are abelian bordered or weakly abelian bordered, the latter meaning that a nonempty proper prefix and a suffix have the same letter frequencies. Theorem 3 shows that if only finitely many factors are weakly abelian unbordered, then the word is bounded weakly abelian periodic, its graph lies in a cylinder with rational axis, and each tangential line to the graph is visited with bounded gaps. Section 7, Question 2 asks whether these three conditions conversely force finitely many weakly abelian unbordered factors; the authors note a positive answer for binary words with letter frequencies one half.

The module D5/S1/Words/AbelianBorders/AbelianBorderQuestion answers Question 2 negatively by an explicit ternary word.

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DOI: 10.48550/arXiv.1501.07464

URL: https://arxiv.org/abs/1501.07464v1

• Locator: Section 7, Question 2; Theorem 3 and the definitions of Sections 2 and 4.