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Avoiding fractional powers over the natural numbers

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Lara Pudwell, Eric Rowland

Combinatorics · Submitted 9 Oct 2015

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

The authors' main claim We identify the structure of the lexicographically least $a/b$-power-free word for three infinite families of rationals $a/b$ as well many "sporadic" rationals that do not seem to belong to general families.

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arXiv ID1510.02807 (version 2)
Pith IDAGOJR4DH, Pith's permanent record of this paper and its review
CitationEric Rowland, Lara Pudwell. "Avoiding fractional powers over the natural numbers." arXiv:1510.02807 (2015). https://pith.science/paper/AGOJR4DH
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,697 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.15 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.92 · compares with prior results 0.12 · mentions code or data availability 0.03.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

We study the lexicographically least infinite $a/b$-power-free word on the alphabet of non-negative integers. Frequently this word is a fixed point of a uniform morphism, or closely related to one. For example, the lexicographically least $7/4$-power-free word is a fixed point of a $50847$-uniform morphism. We identify the structure of the lexicographically least $a/b$-power-free word for three infinite families of rationals $a/b$ as well many "sporadic" rationals that do not seem to belong to general families. To accomplish this, we develop an automated procedure for proving $a/b$-power-freeness for morphisms of a certain form, both for explicit and symbolic rational numbers $a/b$. Finally, we establish a connection to words on a finite alphabet. Namely, the lexicographically least $27/23$-power-free word is in fact a word on the finite alphabet $\{0, 1, 2\}$, and its sequence of letters is $353$-automatic.

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 We study the lexicographically least infinite $a/b$-power-free word on the alphabet of non-negative integers. contribution 0.88 0.90 0.06 asserted 0.93
s1 Frequently this word is a fixed point of a uniform morphism, or closely related to one. result 0.74 0.53 0.06 hedged possibility 0.63
s2 For example, the lexicographically least $7/4$-power-free word is a fixed point of a $50847$-uniform morphism. result 0.98 0.81 0.07 asserted 1.00
s3 We identify the structure of the lexicographically least $a/b$-power-free word for three infinite families of rationals $a/b$ as well many "sporadic" rationals that do not seem to belong to general families. contribution 0.51 0.95 0.51 asserted 0.99
s4 To accomplish this, we develop an automated procedure for proving $a/b$-power-freeness for morphisms of a certain form, both for explicit and symbolic rational numbers $a/b$. method 0.98 0.96 0.07 asserted 1.00
s5 Finally, we establish a connection to words on a finite alphabet. result 0.54 0.90 0.11 asserted 0.97
s6 Namely, the lexicographically least $27/23$-power-free word is in fact a word on the finite alphabet $\{0, 1, 2\}$, and its sequence of letters is $353$-automatic. result 0.99 0.92 0.24 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.66 · evidence from proof or derivation 0.92.

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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

We study the lexicographically least infinite $a/b$-power-free word on the alphabet of non-negative integers. Frequently this word is a fixed point of a uniform morphism, or closely related to one. For example, the lexicographically least $7/4$-power-free word is a fixed point of a $50847$-uniform morphism. We identify the structure of the lexicographically least $a/b$-power-free word for three infinite families of rationals $a/b$ as well many "sporadic" rationals that do not seem to belong to general families. To accomplish this, we develop an automated procedure for proving $a/b$-power-freeness for morphisms of a certain form, both for explicit and symbolic rational numbers $a/b$. Finally, we establish a connection to words on a finite alphabet. Namely, the lexicographically least $27/23$-power-free word is in fact a word on the finite alphabet $\{0, 1, 2\}$, and its sequence of letters is $353$-automatic.

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