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Quantum Kronecker fractions

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S.J. Evans, A.P. Veselov, B. Winn

Number Theory · Submitted 21 Oct 2024

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

The authors' main claim We prove that the corresponding continued fraction expansions must be palindromic and describe all Kronecker fractions with prime denominators.

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arXiv ID2410.15666 (version 2)
Pith ID7SCXU4Z5, Pith's permanent record of this paper and its review
CitationA.P. Veselov, B. Winn, S.J. Evans. "Quantum Kronecker fractions." arXiv:2410.15666 (2024). https://pith.science/paper/7SCXU4Z5
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,243 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.20 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.85 · compares with prior results 0.62 · mentions code or data availability 0.03.

The abstract, sentence by sentence

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

A few years ago Morier-Genoud and Ovsienko introduced an interesting quantization of the real numbers as certain power series in a quantization parameter $q.$ It is known now that the golden ratio has minimal radius among all these series. We study the rational numbers having maximal radius of convergence equal to 1, which we call Kronecker fractions. We prove that the corresponding continued fraction expansions must be palindromic and describe all Kronecker fractions with prime denominators. We found several infinite families of Kronecker fractions and all Kronecker fractions with denominator less than 5000. We also comment on the irrational case and on the relation with braids, rational knots and links.

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 few years ago Morier-Genoud and Ovsienko introduced an interesting quantization of the real numbers as certain power series in a quantization parameter $q.$ It is known now that the golden ratio has minimal radius among all these series. background 1.00 0.04 0.04 asserted 1.00
s1 We study the rational numbers having maximal radius of convergence equal to 1, which we call Kronecker fractions. contribution 0.99 0.74 0.12 asserted 0.94
s2 We prove that the corresponding continued fraction expansions must be palindromic and describe all Kronecker fractions with prime denominators. result 0.91 0.95 0.58 asserted 1.00
s3 We found several infinite families of Kronecker fractions and all Kronecker fractions with denominator less than 5000. result 0.89 0.97 0.16 asserted 1.00
s4 We also comment on the irrational case and on the relation with braids, rational knots and links. contribution 0.52 0.63 0.04 unclear 0.55

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

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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

A few years ago Morier-Genoud and Ovsienko introduced an interesting quantization of the real numbers as certain power series in a quantization parameter $q.$ It is known now that the golden ratio has minimal radius among all these series. We study the rational numbers having maximal radius of convergence equal to 1, which we call Kronecker fractions. We prove that the corresponding continued fraction expansions must be palindromic and describe all Kronecker fractions with prime denominators. We found several infinite families of Kronecker fractions and all Kronecker fractions with denominator less than 5000. We also comment on the irrational case and on the relation with braids, rational knots and links.

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2 arXiv papers cite this in 3 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

What it cites · 3 sentences without a confident label

Compares with 1 paper
  • Corollary [ref] appears as Theorem 3.6 in the recent work [ref] , where it is proved by a different method.

    Arithmetic on $q$-deformed rational numbers · Maximal radius of convergence and Kronecker fractions · 0.88

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