Metric properties of continued fractions with large prime partial quotients

Wanjin Cheng; Wen Wu

arxiv: 2510.27284 · v2 · pith:5JIXHXW7new · submitted 2025-10-31 · 🧮 math.NT

Metric properties of continued fractions with large prime partial quotients

Wanjin Cheng , Wen Wu This is my paper

Pith reviewed 2026-05-21 20:12 UTC · model grok-4.3

classification 🧮 math.NT

keywords continued fractionspartial quotientsprime numbersLebesgue measureHausdorff dimensionzero-one lawmetric number theory

0 comments

The pith

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.

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

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.

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).

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.

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.

Where Pith is reading between the lines

These are 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.

Load-bearing premise

The function φ is non-decreasing.

What would settle it

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

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

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

The main thing to know is that the abstract writes E'(φ) with ∃k≠l≤n before the “for i.m. n” clause. For non-decreasing φ going to infinity this forces the set to be empty, since no fixed partial quotients can exceed larger and larger φ(n) infinitely often. The zero-one law and Hausdorff dimension statements then have nothing to apply to. If the paper actually intends the usual formulation—for infinitely many n there exist distinct k,l ≤n with a'_k and a'_l at least φ(n)—then the results are a direct extension of earlier metric work on large partial quotients, now restricted to primes via the a' notation that zeros out composites. That restriction is cleanly defined and the theorems are stated clearly. The paper does the expected job of carrying over the standard Borel-Cantelli and dimension estimates to this variant, which is the natural next step in the subfield. The proofs are not visible in the abstract, so it is impossible to check whether the prime condition creates extra technical steps or whether the arguments go through without new error terms. The non-decreasing assumption on φ is standard and not a serious limitation here. This is narrow metric number theory aimed at people who already work with continued-fraction limsup sets and Hausdorff dimensions. A reader outside that circle will not find much to use. The work is coherent enough on its own terms to deserve a serious referee who can verify the quantifier order in the body and the details of the estimates. I would send it to review rather than desk-reject.

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)

[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

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

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.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 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.

axioms (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).

pith-pipeline@v0.9.0 · 5684 in / 1417 out tokens · 89400 ms · 2026-05-21T20:12:32.360755+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

E'(φ) := {x : ∃ 1≤k≠l≤n, a'_k(x), a'_l(x) ≥ φ(n) for i.m. n}, zero-one law L(E'(φ))=0/1 according as ∑ n/φ(n)^2 log²φ(n) converges/diverges; dim_H via pressure P(T,f) with f=−(3s−1)log B − s log|T'|
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Proof uses Borel–Cantelli, Chung–Erdős, cylinder length 1/q_n², Prop 2.4 ∑_{p≥M} 1/p² ≍ 1/(M log M)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

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

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

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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 Wu)School of Mathematics, South China University of Technology, Guangzhou, 5106...

work page 2008

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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

[9] [9]

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

work page 1964

[10] [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

work page 1926

[11] [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

work page 2014

[12] [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

work page 1997

[13] [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

work page 1995

[14] [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

work page 1996

[15] [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

work page 1999

[16] [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

work page 2007

[17] [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

work page 1988

[18] [18]

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

work page arXiv 2025

[19] [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

work page 2023

[20] [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

work page 2023

[21] [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 Wu)School of Mathematics, South China University of Technology, Guangzhou, 5106...

work page 2008