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Metric properties of continued fractions with large prime partial quotients

T0 review · 1 major / 0 minor · reviewed 2026-05-21 · grok-4.3

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Pith's one-line read The set E'(φ) of continued fractions with at least two large prime partial quotients infinitely often obeys a zero-one law for Lebesgue measure.

desk verdict The abstract definition of E'(φ) puts the quantifiers in an order that empties the set for typical φ, so the zero-one law and dimension claims are vacuous unless the body reorders them to the standard limsup form. read the letter →

arxiv 2510.27284 v2 pith:5JIXHXW7 submitted 2025-10-31 math.NT

Wanjin Cheng , Wen Wu This is my paper · ORCID
classification math.NT
keywords continuedfractionspartialquotientsprimenumbersLebesguemeasureHausdorffdimensionzero-onelawmetricnumbertheory
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper studies numbers x in [0,1) whose continued fraction expansions contain at least two prime partial quotients that exceed a given non-decreasing function φ(n) at infinitely many stages n. It proves that the Lebesgue measure of the set of all such x is either zero or one. The paper also computes the Hausdorff dimension of this set. A reader would care because the result describes how frequently large prime quotients appear together in continued fraction expansions, which controls the metric properties of Diophantine approximations involving primes.

What carries the argument

The set E'(φ) tracking simultaneous occurrences of two or more large prime partial quotients a'_i(x) ≥ φ(n) for indices up to n, infinitely often, which is the object whose measure and dimension are analyzed via metric arguments on the continued fraction expansion.

What would settle it

An explicit non-decreasing φ for which the Lebesgue measure of E'(φ) lies strictly between 0 and 1.

Watch

Extended reading notes

Core claim

Let φ be a non-decreasing function from the natural numbers to the positive reals. Define E'(φ) as the set of x in [0,1) such that there exist distinct k and l at most n with a'_k(x) and a'_l(x) both at least φ(n) for infinitely many n, where a'_i(x) equals a_i(x) if a_i(x) is prime and equals zero otherwise. The paper establishes a zero-one law for the Lebesgue measure of E'(φ) and determines the Hausdorff dimension of E'(φ).

Load-bearing premise

The function φ is non-decreasing.

Editorial extensions

If this is right

  • The Lebesgue measure of E'(φ) is either 0 or 1 according to a divergence criterion on φ.
  • The Hausdorff dimension of E'(φ) equals an explicit value determined by the growth rate of φ.
  • The zero-one law and dimension formula hold uniformly for all non-decreasing φ.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same approach may apply when the primality condition is replaced by other arithmetic restrictions on the partial quotients.
  • Results of this type could be used to study the distribution of prime denominators in best rational approximations.
  • The zero-one law suggests that the appearance of multiple large prime quotients is governed by the same Borel-Cantelli type phenomena that control ordinary large partial quotients.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 0 minor

Summary. The paper defines the set E'(φ) of x in [0,1) whose continued fraction expansions contain at least two distinct prime partial quotients a'_k(x) and a'_l(x) each at least φ(n) for infinitely many n, where φ is non-decreasing. It claims to prove a zero-one law for the Lebesgue measure of E'(φ) and to compute its Hausdorff dimension.

Significance. If the central claims hold with the intended (limsup) formulation of E'(φ), the results would extend metric theory of continued fractions to the setting of simultaneously large prime partial quotients, providing a zero-one law and dimension formula that could be of interest in Diophantine approximation and geometric measure theory.

major comments (1)
  1. [Abstract / Definition of E'(φ)] Abstract (and presumably §1): the definition of E'(φ) is written with the existential quantifier ∃1≤k≠l≤n placed before 'for i.m. n'. Taken literally this asserts existence of fixed indices k,l (hence fixed a'_k, a'_l) such that a'_k, a'_l ≥ φ(n) for all sufficiently large n. Since φ is non-decreasing and φ(n)→∞, no such fixed finite values exist, so E'(φ) is empty. The zero-one law and Hausdorff-dimension statements are then vacuous. The manuscript must explicitly adopt the limsup formulation (for infinitely many n there exist k≠l≤n with both a'_k,a'_l ≥φ(n)) and verify that all subsequent arguments use this corrected definition. This quantifier order is load-bearing for every stated result.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying the ambiguity in the quantifier order within the definition of E'(φ). We agree that the current notation risks misinterpretation and will revise the manuscript to adopt an explicit limsup formulation.

read point-by-point responses

  1. Referee: Abstract (and presumably §1): the definition of E'(φ) is written with the existential quantifier ∃1≤k≠l≤n placed before 'for i.m. n'. Taken literally this asserts existence of fixed indices k,l (hence fixed a'_k, a'_l) such that a'_k, a'_l ≥ φ(n) for all sufficiently large n. Since φ is non-decreasing and φ(n)→∞, no such fixed finite values exist, so E'(φ) is empty. The zero-one law and Hausdorff-dimension statements are then vacuous. The manuscript must explicitly adopt the limsup formulation (for infinitely many n there exist k≠l≤n with both a'_k,a'_l ≥φ(n)) and verify that all subsequent arguments use this corrected definition. This quantifier order is load-bearing for every stated result.

    Authors: We agree that the notation in the abstract (and the corresponding definition in §1) is ambiguous and does not unambiguously express the intended meaning. The set E'(φ) is meant to consist of those x for which there are infinitely many n such that there exist distinct indices k, l ≤ n with a'_k(x) and a'_l(x) both prime and at least φ(n). We will rewrite the abstract and the definition in §1 to state this limsup formulation explicitly, for example by placing the existential quantifier inside the 'for infinitely many n' clause. We will also review the proofs in §§2–4 to confirm that they already operate under this interpretation (as the metric arguments rely on the existence of such pairs for infinitely many n rather than fixed indices) and will add clarifying remarks or minor adjustments where needed to make the dependence on the corrected definition explicit. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity in derivation; results follow from standard metric theory on the given set definition

full rationale

The paper defines the set E'(φ) explicitly in the abstract and states that it establishes a zero-one law for its Lebesgue measure together with its Hausdorff dimension. These claims rest on the definition of E'(φ) combined with standard tools from continued fractions, Diophantine approximation, and dimension theory. No step reduces a prediction to a fitted parameter by construction, invokes a self-citation as the sole justification for a uniqueness theorem, or renames a known result as a new derivation. The central results therefore remain independent of the inputs once the set is fixed, yielding a self-contained argument.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

Only the abstract is available, so the ledger is limited to explicitly mentioned elements. No free parameters or invented entities are introduced. The work relies on standard background from measure theory and continued fraction dynamics.

assumptions (2)
  • standard math Continued fraction expansions exist and are unique for irrational numbers in [0,1).
    Implicit in the definition of partial quotients a_i(x) and the set E'(φ).
  • standard math Lebesgue measure and Hausdorff dimension are well-defined and applicable to subsets of [0,1).
    Used to state the zero-one law and dimension result for E'(φ).

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Cite this review

Pith. "Pith review of Metric properties of continued fractions with large prime partial quotients." pith.science (2026). https://pith.science/paper/5JIXHXW7

@misc{pith2026251027284,
  author       = {Pith},
  title        = {Pith review of: Metric properties of continued fractions with large prime partial quotients},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5JIXHXW7}},
  note         = {Machine review of arXiv:2510.27284}
}
abstract

Let $x \in [0,1)$ with continued fraction expansion $[a_1(x),a_2(x),\dots]$, and let $\phi:\mathbb{N}\to\mathbb{R}^+$ be a non-decreasing function. We consider the numbers whose continued fraction expansions contain at least two partial quotients that are simultaneously large and prime, that is \[ E'(\phi):=\Big\{x\in[0,1): \exists\, 1\leq k\neq l\leq n, \ a'_{k}(x),\ a'_{l}(x)\geq\phi(n) \ \text{for i.m. } n\in\mathbb{N}\Big\}, \] where $a'_i(x)$ denotes $a_i(x)$ if $a_i(x)$ is prime and $0$ otherwise. We establish a zero-one law for the Lebesgue measure of $E'(\phi)$ and determine its Hausdorff dimension.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

21 extracted references · 21 canonical work pages

  1. [1]

    Bernstein, ¨Uber eine Anwendung der Mengenlehre auf ein aus der Theorie der s¨ akularen St¨ orungen herr¨ uhrendes Problem, Math

    F. Bernstein, ¨Uber eine Anwendung der Mengenlehre auf ein aus der Theorie der s¨ akularen St¨ orungen herr¨ uhrendes Problem, Math. Ann.71(1911), no. 3, 417–439

    work page 1911

  2. [2]

    Bordignon, D

    M. Bordignon, D. R. Johnston and V. Starichkova, An explicit version of Chen’s theorem and the linear sieve, arXiv:2207.09452, 2025

    work page arXiv 2025

  3. [3]

    Borel, Sur un probl` eme de probabilit´ es relatif aux fractions continues, Math

    ´E. Borel, Sur un probl` eme de probabilit´ es relatif aux fractions continues, Math. Ann.72(1912), no. 4, 578–584

    work page 1912

  4. [4]

    K. L. Chung and P. Erd¨ os, On the application of the Borel-Cantelli lemma, Trans. Amer. Math. Soc. 72(1952), 179–186

    work page 1952

  5. [5]

    Dajani and C

    K. Dajani and C. Kraaikamp,Ergodic theory of numbers, Carus Mathematical Monographs, 29, Math. Assoc. America, Washington, DC, 2002

    work page 2002

  6. [6]

    H. G. Diamond and J. D. Vaaler, Estimates for partial sums of continued fraction partial quotients, Pacific J. Math.122(1986), no. 1, 73–82

    work page 1986

  7. [7]

    K. J. Falconer,Fractal geometry, third edition, Wiley, Chichester, 2014

    work page 2014

  8. [8]

    P. G. Hanus, R. D. Mauldin and M. Urba´ nski, Thermodynamic formalism and multifractal analysis of conformal infinite iterated function systems, Acta Math. Hungar.96(2002), no. 1-2, 27–98

    work page 2002

Show all 21 references
  1. [9]

    A. Y. Khinchin,Continued fractions, Univ. Chicago Press, Chicago, Ill.-London, 1964

    1964

  2. [10]

    Knopp, Mengentheoretische Behandlung einiger Probleme der diophantischen Approximationen und der transfiniten Wahrscheinlichkeiten, Math

    K. Knopp, Mengentheoretische Behandlung einiger Probleme der diophantischen Approximationen und der transfiniten Wahrscheinlichkeiten, Math. Ann.95(1926), no. 1, 409–426

    1926

  3. [11]

    B. Li, B. W. Wang, J. Wu and J. Xu, The shrinking target problem in the dynamical system of continued fractions, Proc. Lond. Math. Soc. (3)108(2014), no. 1, 159–186. 18 W. CHENG AND W. WU

    2014

  4. [12]

    Luczak, On the fractional dimension of sets of continued fractions, Mathematika44(1997), no

    T. Luczak, On the fractional dimension of sets of continued fractions, Mathematika44(1997), no. 1, 50–53

    1997

  5. [13]

    Mattila,Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, 44, Cambridge Univ

    P. Mattila,Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, 44, Cambridge Univ. Press, Cambridge, 1995

    1995

  6. [14]

    R. D. Mauldin and M. Urba´ nski, Dimensions and measures in infinite iterated function systems, Proc. London Math. Soc. (3)73(1996), no. 1, 105–154

    1996

  7. [15]

    R. D. Mauldin and M. Urba´ nski, Conformal iterated function systems with applications to the geom- etry of continued fractions, Trans. Amer. Math. Soc.351(1999), no. 12, 4995–5025

    1999

  8. [16]

    H. L. Montgomery and R. C. Vaughan,Multiplicative number theory. I. Classical theory, Cambridge Studies in Advanced Mathematics, 97, Cambridge Univ. Press, Cambridge, 2007

    2007

  9. [17]

    Philipp, Limit theorems for sums of partial quotients of continued fractions, Monatsh

    W. Philipp, Limit theorems for sums of partial quotients of continued fractions, Monatsh. Math.105 (1988), no. 3, 195–206

    1988

  10. [18]

    G. G. Robert, M. Hussain, B. Ward and L. White, Continued fractions with large prime partial quotients, arXiv:2507.17167, 2025

    2025

  11. [19]

    T. I. Schindler and R. Zweim¨ uller, Prime numbers in typical continued fraction expansions, Boll. Unione Mat. Ital.16(2023), no. 2, 259–274

    2023

  12. [20]

    B. Tan, C. Tian and B. W. Wang, The distribution of the large partial quotients in continued fraction expansions, Sci. China Math.66(2023), no. 5, 935–956

    2023

  13. [21]

    B. W. Wang and J. Wu, Hausdorff dimension of certain sets arising in continued fraction expansions, Adv. Math.218(2008), no. 5, 1319–1339. (Wanjin Cheng)School of Mathematics, South China University of Technology, Guangzhou, 510640, China Email address:chengwj0227@163.com (Wen...

    2008

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