bibkey: fazekas2026binary authors: Szilárd Zsolt Fazekas, Adam Mammoliti, Robert Mercaş, Jamie Simpson year: 2026 title: Binary Words Containing Few Abelian Squares doi: 10.48550/arXiv.2604.23188 url: https://arxiv.org/abs/2604.23188v1 claim: Conjecture 4 asserts that equality in the floor n over 4 lower bound forces every abelian square to be trivial. strata_touched:

• D5/S1/Words/Powers/FazekasAbelianSquareEqualityRefutation license: citation-only triage: anchor

Binary Words Containing Few Abelian Squares

Conjecture 4 on printed page 1 states that a binary word of length n contains at least floor(n/4) abelian squares and that, when equality holds, all of them are trivial. An abelian square is a factor whose two equal-length halves have the same Parikh vector. A trivial abelian square is a positive even power of a single letter.

The paper’s example abaababa has the six distinct abelian-square factors aa, abab, abaaba, baba, baab, and aababa. Table 1 likewise counts distinct factors and lists abab and baba in its length-four row.

This note attests the definitions and the published conjecture, not the repository’s refutation. Bounded checks of the arXiv version record, arXiv abstract search, exact title and identifier searches, Semantic Scholar, and Crossref found no later settlement. That search scope is not an exhaustive claim about all literature.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2604.23188
• URL: https://arxiv.org/abs/2604.23188v1
• Version and location: arXiv:2604.23188v1, printed pages 1-2, Conjecture 4 and Table 1.