Pith. sign in

REVIEW 3 major objections 6 minor 44 references

Continued fractions with large prime partial quotients

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper proves that requiring infinitely many partial quotients of a continued fraction to be prime leaves the Hausdorff dimension of the associated sets unchanged, and it determines their Lebesgue measure through a zero-one law driven…

desk verdict Solid extension of Huang-Wu-Xu to prime partial quotients; main theorems hold, with fixable typos and one statement-proof mismatch in Theorem 1.7. read the letter →

arxiv 2507.17167 v1 pith:F4UPL43D submitted 2025-07-23 math.NT math.DS

Gerardo González Robert , Mumtaz Hussain , Benjamin Ward , Lauren White This is my paper · ORCID
classification math.NTmath.DS MSC 11A5511K5028A8011N05
keywords continuedfractionsprimepartialquotientsHausdorffdimensionLebesguemeasurezero-onelawalmostzetafunctionBorel–BernsteintheoremDiophantineapproximation
topics The Twin Prime Conjecture
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 sets of real numbers whose continued-fraction partial quotients are, infinitely often, both large and prime. Its main claim is that this primality restriction does not change the Hausdorff dimension: for every function $\varphi$ and every $\ell$, the dimension of $E'_\ell(\varphi)$ equals that of the unrestricted set $E_\ell(\varphi)$. It also proves a zero-one law for Lebesgue measure, with the threshold series $\sum (\log\log\varphi(n))^{\ell-1}/(\varphi(n)\log\varphi(n))$. The engine behind both results is a new asymptotic for the tail of the almost prime zeta function. If the paper is right, the dimension formulas for large partial quotients carry over verbatim to prime partial quotients, while the Lebesgue-measure threshold changes by replacing one logarithm by a double logarithm.

What carries the argument

The central analytic object is the almost prime zeta tail $P_\ell(s;M)$, the sum of $k^{-s}$ over integers $k\ge M$ having at most $\ell$ prime factors; the asymptotic of Theorem 1.7 converts prime-counting information into interval-measure estimates. The dimension argument is carried by a Cantor-set construction on fundamental intervals whose distinguished blocks are filled by primes in dyadic ranges, with gaps controlled by the explicit short-interval prime bound of Proposition 2.4 and lengths controlled by the denominator recursion for continued fractions. The Mass Distribution Principle then turns H\"older estimates for the constructed measure into the lower bound $s(1-\delta)-\delta$, and letting $\delta\to 0$ and $s\to t_B^{(\ell)}$ yields the equality in Theorem 1.10.

What would settle it

Exhibiting a single $x>e^{20}$ with no prime in $[0.999x,x)$ would falsify the explicit prime-gap input and invalidate the gap estimates behind the dimension lower bound. Short of that, a direct computation of $\dim_H E'_2(\varphi)$ for $\varphi(n)=2^n$ that disagrees with the predicted pressure value $t_2^{(2)}$ would refute Theorem 1.10.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the metric theory of continued fractions with large partial quotients is unchanged when the allowed partial quotients are restricted to primes. Theorem 1.10 states $\dim_H E'_\ell(\varphi)=\dim_H E_\ell(\varphi)$ for every $\varphi:\mathbb{N}\to\mathbb{R}_{>0}$ and $\ell\in\mathbb{N}$, so the full dimension formula of the unrestricted case, expressed through the pressure function $t_B^{(\ell)}$, transfers verbatim to the prime setting. Theorem 1.8 gives $m(E'_\ell(\varphi))=0$ or $1$ according as the series $\sum (\log\log\varphi(n))^{\ell-1}/(\varphi(n)\log\varphi(n))$ converges or diverges, the prime analogue of the Borel-Bernstein-type law in which the factor $\log\varphi(n)$ is replaced by $\log\log\varphi(n)$ and an extra $\log\varphi(n)$ appears in the denominator. The supporting analytic input is Theorem 1.7: for fixed $\ell$ and $s>1$, the tail $P_\ell(s;M)=\sum_{k\in\Omega(\ell),\,k\ge M}k^{-s}$ is comparable to $(\log\log M)^{\ell-1}/(M^{s-1}\log M)$.

Load-bearing premise

The load-bearing premise is that every interval $[0.999x,x)$ with $x>e^{20}$ contains at least one prime; if consecutive primes could be spaced more widely than one tenth of one percent of $x$, the Cantor blocks would have gaps too large for the dimension lower bound to survive.

Editorial extensions

If this is right

  • The Lebesgue measure of $E'_\ell(\varphi)$ is now fully determined for $\varphi:\mathbb{N}\to[3,\infty)$: the set is null or full according to the convergence or divergence of $\sum (\log\log\varphi(n))^{\ell-1}/(\varphi(n)\log\varphi(n))$.
  • Every Hausdorff dimension formula proved for unrestricted large partial quotients, including the value $1/(b+1)$ for exponential thresholds, holds verbatim when the large partial quotients are required to be prime.
  • The zero-one law extends to $F_\ell(\varphi)$, the version where the threshold is evaluated at the denominator $q_n$ of the convergent rather than at $n$.
  • Sets with exponentially growing blocks of prime partial quotients have the same Hausdorff dimension as their unrestricted counterparts, as stated in Proposition 1.12.
  • The tail asymptotic for the almost prime zeta function at $s=2$ is the quantitative reason the measure law takes the form it does, and it is a self-contained number-theoretic byproduct.

Reading between the lines

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

  • A natural extension, not pursued here, would replace the primes by any sparse set $A$ whose counting function grows regularly, such as squares or powers of $2$; the same mechanism would predict dimension equality whenever every interval $[x,(1+\varepsilon)x]$ eventually contains an element of $A$.
  • The contrast between the measure thresholds in the unrestricted and prime cases suggests a heuristic reading: Hausdorff dimension is blind to a $1/\log p$ density factor, whereas Lebesgue measure is not; this interpretation is an editorial gloss, not a claim of the paper.
  • A testable intermediate step is to prove the analogue of Proposition 1.12 when the positions of the large prime quotients are prescribed by a sparse infinite set $\mathcal{N}$; the flexible-block lemma of Section 6.5 already contains most of the technology needed.
  • Because the proof leans only on a short-interval prime bound with gaps of size $O(x)$ and on the prime-counting rate from the Prime Number Theorem, the dimension equality would likely survive for any allowed-quotient set with the same two features, making primality one instance of a broader phenomenon.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper studies the metric theory of continued fractions whose partial quotients are restricted to primes and are required to be large. For functions φ: N → R_{>0}, it defines E'_ℓ(φ) as the set of x for which a product of ℓ consecutive prime partial quotients is at least φ(n) infinitely often, and E''_ℓ(φ) for the analogous condition holding for all n. The main results are a zero-one law for the Lebesgue measure of E'_ℓ(φ) (Theorem 1.8), an equality of Hausdorff dimensions dim_H E'_ℓ(φ) = dim_H E_ℓ(φ) with the unrestricted set (Theorem 1.10), a Łuczak-type dimension theorem for E''_ℓ and E'_ℓ in the double-exponential case (Theorem 1.11), and an asymptotic for the tail of the almost prime zeta function (Theorem 1.7). The proofs follow the Cantor-set and pressure-function framework of Huang-Wu-Xu, replacing counts over all integers by counts over primes via the Prime Number Theorem, Mertens' theorem, and an explicit prime-gap result.

Significance. If the results hold, Theorem 1.10 is a clean and natural statement: imposing primality on the large partial quotients does not change the Hausdorff dimension, and Theorem 1.8 gives the exact prime analogue of the Huang-Wu-Xu zero-one law. The paper is careful and constructive: the Cantor-set lower bound is built explicitly, the estimates are traced back to standard number-theoretic inputs, and there are no fitted parameters or circular reductions. The almost prime zeta tail estimate of Theorem 1.7 is also of independent interest. However, the written proof contains statement/proof mismatches, most notably a possible gap in the central finite-B_φ case of Theorem 1.10 and an incomplete reconciliation between the definition of Ω(ℓ) and the quantity actually estimated in Section 3.

major comments (3)
  1. [Section 3, Theorem 1.7] The theorem is stated for P_ℓ(s;M) with Ω(ℓ) defined as the set of positive integers with 'at most ℓ' prime factors, but the proof establishes the estimate only for S(ℓ,M,1,s), the ordered sum over products of exactly ℓ primes. These two objects are not identical, and the passage between them is not written. Please either redefine Ω(ℓ) as the set of integers with exactly ℓ (not necessarily distinct) prime factors and add the comparison S(ℓ,M,1,s) ≍ ∑_{Ω(n)=ℓ, n≥M} n^{-s}, or, if 'at most ℓ' is intended, add the argument that the sums over products of fewer than ℓ primes are of strictly smaller order and do not affect the asymptotic. Since Theorem 1.8 invokes Theorem 1.7, this point must be made explicit.
  2. [Section 6.6, proof of Theorem 1.10] The written proof of Theorem 1.10 only discusses the cases B_φ = 1 and B_φ = +∞. The case 1 < B_φ < ∞, which is precisely the case prepared by the construction in Sections 6.1-6.5, is missing: the paragraph beginning 'First, assume that B_φ = 1' runs only for C > 1, and the next paragraph jumps to B_φ = +∞. The natural repair is to run the same argument for any finite B_φ > 1, defining N_C = {n : log φ(n)/n < log C} for C > B_φ, using E'_ℓ(φ_C; N_C) ⊆ E'_ℓ(φ), the equality dim_H E'_ℓ(φ_C; N_C) = dim_H E'_ℓ(φ_C) from Lemma 6.6, and then letting C ↓ B_φ. This is routine, but as it stands the central theorem has a gap in its main case and the section needs to be rewritten.
  3. [Section 6.6, B_φ = +∞ subcase] The inclusion E''_ℓ(φ_{e,b_φ+δ}) ⊆ E'_ℓ(φ) is asserted without proof. It is true, but the justification is not immediate from the displayed limit comparison: one must observe that liminf_n (log log φ(n))/n = log b_φ < log(b_φ + δ) implies that φ_{e,b_φ+δ}(n) ≤ φ(n) for infinitely many n, so any x whose all-block product exceeds the faster-growing function still satisfies the infinitely-often condition for φ. Please supply this argument. In addition, the case b_φ < 1 is not discussed; if it is excluded by the earlier reduction to Lebesgue-null sets, that reduction should be stated explicitly.
minor comments (6)
  1. [Proposition 2.3] The displayed formula appears to have the fraction inverted: it should read #(P ∩ [γ^n, 2γ^n]) = c_n(γ) γ^n / (n log γ), since the Prime Number Theorem gives a count asymptotic to γ^n / (n log γ). With the printed formula, c_n(γ) → 1 would force the count to tend to 0, contradicting the intended meaning and the later usage in Section 6.
  2. [Section 3, proof of Theorem 1.7] The base case is introduced with 'Consider an arbitrary s > 2', but the argument via Karamata's theorem works for every s > 1. The statement of Theorem 1.7 also says s > 1, so the proof should be adjusted to s > 1 or the statement should be restricted.
  3. [Introduction, Proposition 1.12] Proposition 1.12 is stated as a result in the introduction, but no proof appears anywhere in the paper, and the surrounding text contains two different definitions labelled S'_ℓ(A_0, ..., A_{ℓ-1}) as well as an undefined S_ℓ. The authors should either prove Proposition 1.12, provide a reference for it, or reformulate it as a remark with a proof sketch; as written, it is an unsupported claim.
  4. [Notation throughout] The notation for the Łuczak-type functions is inconsistent: the introduction uses φ_{b,c}, while Theorem 1.11 and Section 5 use φ_{c,b}; Section 6.6 also uses φ_{e,b_φ+δ} in a way that mixes the two conventions. Please unify the notation.
  5. [Section 6.1.1, condition (24)] The inequalities in condition (24) contain several parentheses and index expressions that are hard to parse, for example '2ℓN ℓ log α0⋯ log αi−2 (α0(α1⋯αi)2αi+1)sδN'. Please rewrite the condition cleanly and verify the indices, since the equality claimed immediately after it is used to justify the measure estimates.
  6. [Section 6.6] There is a spelling error: 'infinte' should read 'infinite'. Also, in the B_φ = 1 paragraph, the phrase 'letting C → B_φ' only makes sense after the correction described in the major comments; in the current text B_φ = 1 and C → 1 is what is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the prime-restricted dimension and measure theorems are derived from PNT, Mertens, and external results rather than from their own conclusions.

full rationale

The paper's central claims are obtained by a forward derivation chain. Theorem 1.7 is proved in Section 3 from the Prime Number Theorem, Karamata's theorem, and Mertens' theorem; Theorem 1.8 then uses Theorem 1.7 at s=2 to estimate m(B_n) and applies the external zero-one law [27, Lemma 3.5]; using one's own earlier theorem inside a proof is not circular because the theorem is proven independently. Theorem 1.10 is proved in Section 6 by constructing a Cantor set whose measure satisfies a Holder bound with exponent approaching the pressure root t_B^(ell) from Huang-Wu-Xu [21], and the lower bound is then matched to dimH E_ell(phi_B) by the already-available Theorem 1.6; the primality content is supplied by Proposition 2.4 (explicit prime in [0.999x,x), external) and Proposition 2.3 (PNT), not by the conclusion. The paper's citations to its own authors' work ([20], [22], [23], [24]) appear in motivational context or side remarks (Proposition 1.12) and are not load-bearing for the main theorems. The proof-technique citations to [21] (Lemmas 6.1 and 6.3, Subsection 5.4.4) are borrowings of an independent unrestricted result, not circular inputs. Minor exposition issues (Theorem 1.7's proof states the base case only for s>2 while the statement claims s>1; the definition of Omega(ell) reads 'at most ell' while the proof treats exactly ell prime factors; Section 6.6 says 'First assume B_phi = 1' where B_phi < infinity is intended; Corollary 1.9's proof is omitted and deferred to [21]) are correctness/exposition risks, but none equates a prediction to a fitted parameter or reduces a theorem to a self-citation. There is no fitted parameter in the paper, and the dimension equality in Theorem 1.10 is established by a construction that attains the known unrestricted dimension from below.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no fitted parameters and no new postulated entities. All external inputs are standard number-theoretic theorems or cited results. The key load-bearing external input is the explicit prime-gap result (Proposition 2.4), which is a recent but plausible theorem.

assumptions (6)
  • standard math Prime Number Theorem with explicit forms, including Rosser's bounds (Theorem 2.5)
    Used to estimate π(x) and prime counts in intervals, in particular Proposition 2.3 and the base case of Theorem 1.7.
  • standard math Mertens' Second Theorem (Theorem 2.1)
    Used in the inductive step of Theorem 1.7 to estimate the sum of 1/p over primes up to M^{1/(ℓ+1)}.
  • standard math Existence of a prime in every interval [0.999x, x) for x > e^{20} (Bordignon-Johnston-Starichkova, Proposition 2.4)
    Used in Lemma 6.2 to ensure fundamental intervals corresponding to consecutive primes are separated and that the union over primes has diameter comparable to the full interval; this is load-bearing for the lower bound in Theorem 1.10.
  • standard math Kleinbock-Wadleigh zero-one law (Lemma 4.1)
    Used to convert measure estimates for B_n into the zero-one law for E'_ℓ(φ) in Theorem 1.8.
  • standard math Huang-Wu-Xu theorems for unrestricted products (Theorems 1.4 and 1.6)
    Used as the benchmark for measure and dimension in the unrestricted case; Theorem 1.10 compares to these.
  • standard math Properties of the pressure function and t_B^(ℓ) (Proposition 2.15 and related results from [21])
    Used to express Hausdorff dimension of E_ℓ(φ) and to justify the approximation as B approaches B_φ.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Continued fractions with large prime partial quotients." pith.science (2026). https://pith.science/paper/F4UPL43D

@misc{pith2026250717167,
  author       = {Pith},
  title        = {Pith review of: Continued fractions with large prime partial quotients},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F4UPL43D}},
  note         = {Machine review of arXiv:2507.17167}
}
read the original abstract

We determine the Lebesgue measure and Hausdorff dimension of various sets of real numbers with infinitely many partial quotients that are both large and prime, thus extending the well-known theorems by {\L}uczak (1997) and Huang-Wu-Xu (2020). To this end, we obtain new asymptotics on the tail end of the almost prime zeta function. Our results include some recent work by Schindler-Zweim{\"u}ller (2023).

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

44 extracted references · 43 canonical work pages

  1. [1]

    R., AND STARICHKOVA , V

    B ORDIGNON , M., J OHNSTON , D. R., AND STARICHKOVA , V. An explicit version of Chen’s theorem. arXiv:2207.09452 (preprint 2025)

    arXiv 2025

  2. [2]

    Measure theoretic properties of large products of consecutive partial quotients

    B ROWN -S ARRE , A., G ONZALEZ ROBERT , G., AND HUSSAIN , M. Measure theoretic properties of large products of consecutive partial quotients. arXiv:2405.10538 (preprint 2025)

    work page Pith review arXiv 2025

  3. [3]

    Approximation by algebraic numbers, vol

    B UGEAUD , Y. Approximation by algebraic numbers, vol. 160 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2004

    work page 2004

  4. [4]

    On the dimension spectrum of infinite subsystems of continued fractions

    C HOUSIONIS , V., L EYKEKHMAN , D., AND URBA ´NSKI , M. On the dimension spectrum of infinite subsystems of continued fractions. Trans. Amer. Math. Soc. 373, 2 (2020), 1009–1042

    work page 2020

  5. [5]

    C USICK , T. W. Sums and products of continued fractions. Proc. Amer. Math. Soc. 27 (1971), 35–38

    work page 1971

  6. [6]

    D AS, T., AND SIMMONS , D. S. Exact dimension functions of the prime continued fraction Cantor set. Ergodic Theory Dynam. Systems 45, 6 (2025), 1757–1776

    work page 2025

  7. [7]

    D AVENPORT , H., AND SCHMIDT , W. M. Dirichlet’s theorem on diophantine approximation. In Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69). Academic Press, London-New York, 1970, pp. 113–132. CONTINUED FRACTIONS WITH LARGE PRIME PARTIAL QUOTIENTS 23

    work page 1968

  8. [8]

    G., AND VAALER , J

    D IAMOND , H. G., AND VAALER , J. D. Estimates for partial sums of continued fraction partial quotients. Pacific J. Math. 122, 1 (1986), 73–82

    work page 1986

Show all 44 references
  1. [9]

    E DWARDS , H. M. Riemann’s zeta function , vol. V ol. 58 of Pure and Applied Mathematics . Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974

    1974

  2. [10]

    Ergodic theory with a view towards number theory, vol

    E INSIEDLER , M., AND WARD , T. Ergodic theory with a view towards number theory, vol. 259 ofGraduate Texts in Mathematics. Springer-Verlag London, Ltd., London, 2011

    2011

  3. [11]

    Fractal geometry, third ed

    F ALCONER , K. Fractal geometry, third ed. John Wiley & Sons, Ltd., Chichester, 2014. Mathematical foundations and applica- tions

    2014

  4. [12]

    On Khintchine exponents and Lyapunov exponents of continued frac- tions

    F AN, A.-H., L IAO, L.-M., W ANG , B.-W., AND WU, J. On Khintchine exponents and Lyapunov exponents of continued frac- tions. Ergodic Theory Dynam. Systems 29, 1 (2009), 73–109

    2009

  5. [13]

    On the fractional dimension of sets of continued fractions

    F ENG , D. J., W U, J., L IANG , J.-C., AND TSENG , S. Appendix to the paper by T. Łuczak—a simple proof of the lower bound: “On the fractional dimension of sets of continued fractions”. Mathematika 44, 1 (1997), 54–55

    1997

  6. [14]

    On the prime zeta function

    F R ¨OBERG , C.-E. On the prime zeta function. Nordisk Tidskr. Informationsbehandling (BIT) 8(1968), 187–202

    1968

  7. [15]

    G LAISHER , J. W. L. On the sums of inverse powers of the prime numbers. Quart. J. Math. 25, 1 (1891), 347–362

  8. [16]

    H ANˇ CL, J., AND TUREK , O. R. Continued fractions with bounded even-order partial quotients. Ramanujan J. 62 , 1 (2023), 69–110

    2023

  9. [17]

    H., AND WRIGHT , E

    H ARDY, G. H., AND WRIGHT , E. M. An introduction to the theory of numbers, fifth ed. The Clarendon Press, Oxford University Press, New York, 1979

    1979

  10. [18]

    Continued fraction Cantor sets, Hausdorff dimension, and functional analysis

    H ENSLEY , D. Continued fraction Cantor sets, Hausdorff dimension, and functional analysis. J. Number Theory 40 , 3 (1992), 336–358

    1992

  11. [19]

    Continued fractions

    H ENSLEY , D. Continued fractions. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006

    2006

  12. [20]

    Limit theorems for sums of products of consecutive partial quotients of continued fractions

    H U, H., H USSAIN , M., AND YU, Y. Limit theorems for sums of products of consecutive partial quotients of continued fractions. Nonlinearity 34, 12 (2021), 8143–8173

    2021

  13. [21]

    Metric properties of the product of consecutive partial quotients in continued fractions

    H UANG , L., W U, J., AND XU, J. Metric properties of the product of consecutive partial quotients in continued fractions. Israel J. Math. 238, 2 (2020), 901–943

    2020

  14. [22]

    Hausdorff dimension analysis of sets with the product of consecutive vs single partial quotients in continued fractions

    H USSAIN , M., L I, B., AND SHULGA , N. Hausdorff dimension analysis of sets with the product of consecutive vs single partial quotients in continued fractions. Discrete Contin. Dyn. Syst. 44, 1 (2024), 154–181

    2024

  15. [23]

    Metrical properties of finite product of partial quotients in arithmetic progressions

    H USSAIN , M., AND SHULGA , N. Metrical properties of finite product of partial quotients in arithmetic progressions. In Press: Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) (2024)

    2024

  16. [24]

    Metrical properties of exponentially growing partial quotients

    H USSAIN , M., AND SHULGA , N. Metrical properties of exponentially growing partial quotients. Forum Math. 37, 5 (2025), 1379–1399

    2025

  17. [25]

    K HINCHIN , A. Y. Continued fractions. University of Chicago Press, Chicago, Ill.-London, 1964

    1964

  18. [26]

    Metrische Kettenbruchprobleme

    K HINTCHINE , A. Metrische Kettenbruchprobleme. Compositio Math. 1 (1935), 361–382

    1935

  19. [27]

    A zero-one law for improvements to Dirichlet’s Theorem

    K LEINBOCK , D., AND WADLEIGH , N. A zero-one law for improvements to Dirichlet’s Theorem. Proc. Amer. Math. Soc. 146, 5 (2018), 1833–1844

    2018

  20. [28]

    The shrinking target problem in the dynamical system of continued fractions

    L I, B., W ANG , B.-W., W U, J., AND XU, J. The shrinking target problem in the dynamical system of continued fractions. Proc. Lond. Math. Soc. (3) 108, 1 (2014), 159–186

    2014

  21. [29]

    On the fractional dimension of sets of continued fractions

    Ł UCZAK , T. On the fractional dimension of sets of continued fractions. Mathematika 44, 1 (1997), 50–53

    1997

  22. [30]

    D., AND URBA ´NSKI , M

    M AULDIN , R. D., AND URBA ´NSKI , M. Conformal iterated function systems with applications to the geometry of continued fractions. Trans. Amer. Math. Soc. 351, 12 (1999), 4995–5025

    1999

  23. [31]

    D., AND URBA ´NSKI , M

    M AULDIN , R. D., AND URBA ´NSKI , M. Graph directed Markov systems, vol. 148 of Cambridge Tracts in Mathematics. Cam- bridge University Press, Cambridge, 2003. Geometry and dynamics of limit sets

    2003

  24. [32]

    L., AND VAUGHAN , R

    M ONTGOMERY , H. L., AND VAUGHAN , R. C. Multiplicative number theory. I. Classical theory, vol. 97 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2007

    2007

  25. [33]

    Limit theorems for sums of partial quotients of continued fractions

    P HILIPP , W. Limit theorems for sums of partial quotients of continued fractions. Monatsh. Math. 105, 3 (1988), 195–206

    1988

  26. [34]

    R ESNICK , S. I. Extreme values, regular variation, and point processes , vol. 4 of Applied Probability. A Series of the Applied Probability Trust. Springer-Verlag, New York, 1987

    1987

  27. [35]

    R ESNICK , S. I. A probability path. Modern Birkh¨auser Classics. Birkh¨auser/Springer, New York, 2014. Reprint of the fifth (2005) printing of the 1999 original [MR1664717]

    2005

  28. [36]

    Explicit bounds for some functions of prime numbers

    R OSSER , B. Explicit bounds for some functions of prime numbers. Amer. J. Math. 63(1941), 211–232

    1941

  29. [37]

    I., AND ZWEIM ¨ULLER , R

    S CHINDLER , T. I., AND ZWEIM ¨ULLER , R. Prime numbers in typical continued fraction expansions. Boll. Unione Mat. Ital. 16, 2 (2023), 259–274

    2023

  30. [38]

    The distribution of the large partial quotients in continued fraction expansions

    T AN, B., T IAN , C., AND WANG , B. The distribution of the large partial quotients in continued fraction expansions. Sci. China Math. 66, 5 (2023), 935–956

    2023

  31. [39]

    Metrical properties of the large products of partial quotients in continued fractions

    T AN, B., AND ZHOU , Q.-L. Metrical properties of the large products of partial quotients in continued fractions. Nonlinearity 37, 2 (2024), Paper No. 025008, 28

    2024

  32. [40]

    T ITCHMARSH , E. C. The theory of the Riemann zeta-function , second ed. The Clarendon Press, Oxford University Press, New York, 1986. Edited and with a preface by D. R. Heath-Brown

    1986

  33. [41]

    Hausdorff dimension of certain sets arising in continued fraction expansions

    W ANG , B.-W., AND WU, J. Hausdorff dimension of certain sets arising in continued fraction expansions. Adv. Math. 218, 5 (2008), 1319–1339

    2008

  34. [42]

    A generalization of the Jarn ´ık-Besicovitch theorem by continued fractions

    W ANG , B.-W., W U, J., AND XU, J. A generalization of the Jarn ´ık-Besicovitch theorem by continued fractions. Ergodic Theory Dynam. Systems 36, 4 (2016), 1278–1306

    2016

  35. [43]

    A remark on the growth of the denominators of convergents

    W U, J. A remark on the growth of the denominators of convergents. Monatsh. Math. 147, 3 (2006), 259–264

    2006

  36. [44]

    Some lacunarity properties of partial quotients of real numbers

    Z HAO, X., AND ZHANG , Z. Some lacunarity properties of partial quotients of real numbers. C. R. Math. Acad. Sci. Paris 362 (2024), 1089–1095. 24 G. GONZ ´ALEZ ROBERT, M. HUSSAIN, B. W ARD, AND L. WHITE DEPARTMENT OF MATHEMATICAL AND PHYSICAL SCIENCES , LA TROBE UNIVERSITY , B...

    2024

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.