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Critical factorisation in square-free words

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Tero Harju

Combinatorics · Submitted 20 Jul 2021

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We show that the sufficiently long square-free words $w$ satisfy $\eta(w) \le |w|-5$ where $|w|$ denotes the length of $w$.

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arXiv ID2107.09421 (version 1)
Pith IDRZQHGQHO, Pith's permanent record of this paper and its review
CitationTero Harju. "Critical factorisation in square-free words." arXiv:2107.09421 (2021). https://pith.science/paper/RZQHGQHO
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,898 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.19 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.99 · compares with prior results 0.31 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

A position $p$ in a word $w$ is critical if the minimal local period at $p$ is equal to the global period of $w$. According to the Critical Factorisation Theorem all words of length at least two have a critical point. We study the number $\eta(w)$ of critical points of square-free ternary words $w$, i.e., words over a three letter alphabet. We show that the sufficiently long square-free words $w$ satisfy $\eta(w) \le |w|-5$ where $|w|$ denotes the length of $w$. Moreover, the bound $|w|-5$ is reached by infinitely many words. On the other hand, every square-free word $w$ has at least $|w|/4$ critical points, and there is a sequence of these words closing to this bound.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 A position $p$ in a word $w$ is critical if the minimal local period at $p$ is equal to the global period of $w$. background 1.00 0.08 0.02 asserted 0.98
s1 According to the Critical Factorisation Theorem all words of length at least two have a critical point. background 1.00 0.03 0.03 asserted 1.00
s2 We study the number $\eta(w)$ of critical points of square-free ternary words $w$, i.e., words over a three letter alphabet. contribution 0.98 0.71 0.05 asserted 0.81
s3 We show that the sufficiently long square-free words $w$ satisfy $\eta(w) \le |w|-5$ where $|w|$ denotes the length of $w$. result 1.00 0.95 0.61 asserted 1.00
s4 Moreover, the bound $|w|-5$ is reached by infinitely many words. result 1.00 0.93 0.19 asserted 1.00
s5 On the other hand, every square-free word $w$ has at least $|w|/4$ critical points, and there is a sequence of these words closing to this bound. result 1.00 0.92 0.14 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

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  • full text
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  • references and citing sentences
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  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

A position $p$ in a word $w$ is critical if the minimal local period at $p$ is equal to the global period of $w$. According to the Critical Factorisation Theorem all words of length at least two have a critical point. We study the number $\eta(w)$ of critical points of square-free ternary words $w$, i.e., words over a three letter alphabet. We show that the sufficiently long square-free words $w$ satisfy $\eta(w) \le |w|-5$ where $|w|$ denotes the length of $w$. Moreover, the bound $|w|-5$ is reached by infinitely many words. On the other hand, every square-free word $w$ has at least $|w|/4$ critical points, and there is a sequence of these words closing to this bound.

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