A century of metric Diophantine approximation and half a decade since Koukoulopoulos-Maynard

Manuel Hauke

REVIEW 2 major objections 6 minor 65 references

A century of metric Diophantine approximation and half a decade since Koukoulopoulos-Maynard

T0 review · 2 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read The note claims that the Duffin-Schaeffer conjecture, central to metric Diophantine approximation, reduces to a single weighted graph bound and a short step on the anatomy of integers, presenting a streamlined proof sketch that extends to…

desk verdict A readable historical survey of metric Diophantine approximation, but the advertised simplified Duffin–Schaeffer proof sketch is not self-contained and the key reduction is misstated as printed. read the letter →

arxiv 2505.08901 v1 pith:KLBLA4OL submitted 2025-05-13 math.NT

Manuel Hauke This is my paper · ORCID

classification math.NT MSC 11J8311K6011N35

keywords metricDiophantineapproximationDuffin-SchaefferconjectureKhintchine'stheoremweightedgraphboundGCDsumssievetheoryBorel-Cantellilemmap-adicvaluations

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 note reviews a century of metric Diophantine approximation, from Khintchine's 1924 theorem on the measure of sets of well-approximable numbers to the 2020 resolution of the Duffin-Schaeffer conjecture for arbitrary approximation functions. Its own contribution is a simplified proof sketch that reduces the Duffin-Schaeffer conjecture to a single weighted graph inequality, and it argues that this simplification is a reusable template for other number-theoretic problems. The sketch also packages the two auxiliary ingredients—a sieve estimate for overlaps and an 'anatomy of integers' count—so that a sympathetic reader can follow the overall strategy without the full machinery of the original proof. The payoff would be a shorter, stronger route to one of the central results in metric Diophantine approximation, with quantitative refinements and extensions built on the same foundation.

What carries the argument

The carrying object is the paper's 'Main technical result' (Theorem 1), a weighted graph bound. For finitely supported weight functions $\psi,\theta$ and any set of pairs $(v,w)$ satisfying $D_{\psi,\theta}(v,w)\le 1$ and $\sum_{p\ge t,\, p|vw/\gcd(v,w)^2} 1/p \ge C$, it asserts $\mu_{\psi,\theta}(E) \le 1000\, P_{\psi,\theta}(\varepsilon)\, (\mu_\psi(V)\mu_\theta(W) e^{-Ct})^{1/2+\varepsilon}$. This bound packages the minimal-counterexample structure from Proposition 2.4 and the anatomy lemmas from Lemmas 2.7 and 2.8 into one inequality; summing it over dyadic scales yields the variance estimate (4), which, by the refined Borel–Cantelli lemma of Theorem 2, gives full measure for the limsup set and hence the Duffin-Schaeffer conjecture.

What would settle it

A finite computer search over squarefree-supported weights $\psi,\theta$ on integer intervals $[X,Y]$, comparing the left-hand side of (17) with the claimed right-hand side for a range of $t$, $C$, and $\varepsilon$ in $(0,2/5]$, would test Theorem 1 directly; a single violation would disprove the main technical result and with it the proof sketch's derivation of the Duffin-Schaeffer conjecture.

Watch

Extended reading notes

Core claim

The central claim is that the Duffin-Schaeffer conjecture—full measure or zero measure for the limsup set of reduced-fraction approximants according as $\sum_q \varphi(q)\psi(q)/q$ diverges or converges—follows from a single variance estimate (4), and that this estimate in turn follows from a combinatorial proposition (Proposition 2.3) about pairs of integers whose large common prime divisors are unusually frequent. The proof sketch shows that a minimal counterexample to that proposition must concentrate its weight on pairs $(v,w)$ whose prime factorizations are almost aligned with one integer $N$, in the sense that $|\nu_p(v/N)|+|\nu_p(w/N)| \le 1$ for every prime $p$. Once this structure is established, two lemmas on the anatomy of integers (Lemmas 2.7 and 2.8) show that the assumed counterexample cannot exist. The paper presents the resulting argument as a simplified, shortened, and strengthened version of the 2020 proof, obtained without the GCD-graph induction of the original.

Load-bearing premise

The derivation rests on Theorem 1, the general weighted graph bound, whose proof is only sketched and depends on two lemmas (2.7 and 2.8) whose proofs are omitted; if that bound is false, the presented reduction to the variance estimate and the Duffin-Schaeffer conjecture would not go through.

Editorial extensions

If this is right

If the sketch is correct, the Duffin-Schaeffer theorem follows from one bilinear graph inequality plus two standard analytic-number-theory lemmas, bypassing the GCD-graph induction of the original proof.
The same framework yields quantitative versions of the conjecture: the method supports a strong law of large numbers for the count of approximating reduced fractions, as in the quantitative results cited in the note.
The minimal-counterexample structure—concentration of weight on pairs with $|\nu_p(v/N)|+|\nu_p(w/N)|\le 1$—is reusable; the note states it has already been adapted to other combinatorial problems in number theory.
Because the Borel-Cantelli step works for doubling measures (Theorem 2), the strategy extends to inhomogeneous settings and to metric spaces beyond the torus whenever the equidistribution condition (12) holds.

Reading between the lines

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

If Theorem 1 can be proved with explicit constants, the Duffin-Schaeffer landscape would be reorganized: future work could optimize the exponent and the constant rather than reconstruct GCD-graph-specific arguments.
The bound's exponential factor $e^{-Ct}$ and exponent $1/2+\varepsilon$ suggest a large-deviation interpretation for the bilinear weight; one could attempt a probabilistic proof of (17) via concentration of measure on pairs drawn from $\mu_\psi\otimes\mu_\theta$, possibly generalizing beyond divisibility structure.
Specializing Theorem 1 to $\theta=\psi$ on a critical interval where $\sum \lambda(A_q)\approx 1$ should yield an explicit rate at which $\lambda(\limsup A_q)$ approaches 1, comparable with the quantitative bounds already in the literature.
The squarefree assumption in the sketch leaves open whether the same graph bound handles higher p-adic valuations; testing whether Lemma 2.8 has an analogue with powers of primes could determine whether the simplified proof extends unchanged to all denominators.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Punchline: this is a readable historical survey of Khintchine and Duffin–Schaeffer, but the advertised simplified proof sketch is not self-contained and the key reduction is misstated as printed. The survey half is the real value.

What is new: no new theorems; the value is expository. The historical narrative is accurate and well organized: it covers Khintchine’s theorem, the role of monotonicity, Duffin–Schaeffer’s counterexample, Gallagher’s 0-1 law, the Koukoulopoulos–Maynard breakthrough, and the main generalizations (manifolds, fractals, inhomogeneous, multiplicative). The bibliography is useful and the level is right for a non-specialist. The prose is clear.

Soft spots: In Section 2, Proposition 2.2 claims a bound on the sum of λ(Aq)λ(Ar), which under the hypothesis [1,2] is trivially at most 4. The reduction to the variance estimate (4) needs the analogous bound for λ(Aq ∩ Ar). The final display in Section 2.3.1 repeats the product-sum conclusion, so the printed chain does not actually derive (4). This is a correctable typo, but it must be fixed before anyone can follow the argument.

More substantively, the heart of the claimed simplified proof is Theorem 1, stated without proof. The proof of Proposition 2.4 outsources its key concentration step to [36, Lemma 3.2] and [30, Lemma 2.1]; Lemmas 2.7 and 2.8 are stated without proof, with pointers to [1] and [36]. Since [36] is an unpublished manuscript, a reader cannot verify the central claim that the proof is simplified. This is not circular — the Duffin–Schaeffer theorem is externally established — but the self-citation is load-bearing and inaccessible. The author should either include the missing arguments or clearly label the sketch as provisional.

The citation pattern is otherwise fine: the standard references are present, and the historical claims match my knowledge. The proof sketch, if corrected, is a useful outline for someone who already knows the K-M strategy, but the paper overclaims self-containedness.

Bottom line: worth reading as a historical survey and as an entry point to the area; not as a self-contained proof. A serious referee would catch the Proposition 2.2 issue and ask for the outsourced steps to be either proved or flagged. For an expository venue (IMN is the stated target), I would send it to peer review with the expectation of revision; for a research journal I would desk-reject, since there is no new mathematics.

Referee Report

2 major / 6 minor

Summary. The paper is a survey article on metric Diophantine approximation, centered on Khintchine's theorem (1924) and on the Duffin–Schaeffer conjecture, which was proved by Koukoulopoulos and Maynard in 2020. The first half gives a historical overview and discusses generalizations (inhomogeneous, on fractals, on manifolds, multiplicative, restricted denominators) as well as probabilistic refinements such as strong laws, central limit theorems, and laws of the iterated logarithm. The second half presents a 'simplified proof sketch' of the Duffin–Schaeffer conjecture, attributed to the author's work with Saez and Walker [36]. The sketch reduces the conjecture to a variance estimate (4), derives an overlap estimate via sieving (Section 2.2), and reduces the core difficulty to a graph-theoretic combinatorial statement (Proposition 2.3, Theorem 1) about the non-existence of sets with only large GCDs. The final part outlines a minimal-counterexample argument producing a structured subgraph (Proposition 2.4) and a contradiction (Proposition 2.5). The paper also claims that the method can be applied in other areas of number theory.

Significance. The historical portion is accurate and well-referenced; the paper correctly explains the role of the Duffin–Schaeffer counterexample, the failure of pointwise overlap bounds, and the need for averaged combinatorial control of GCD sums. It also collects recent developments, including several 2024–2025 preprints, which makes it a useful entry point to the literature. If the simplified proof sketch is correct, it would be a valuable exposition of a major theorem and could indeed serve as a template for other problems, as the paper argues. However, the proof sketch as printed is not self-contained and currently does not establish the advertised claim: Proposition 2.2 misstates the needed estimate, and the central combinatorial results (Theorem 1, Lemmas 2.7–2.8) are not proved but delegated to an unpublished manuscript [36]. The paper's value therefore depends on external results that a reader cannot currently check. The exposition is promising, but the mathematical content must be made verifiable before the paper can be accepted.

major comments (2)

[Section 2.1, Proposition 2.2 and Section 2.3.1] Proposition 2.2 states a bound on ∑_{X≤q,r≤Y} λ(A_q)λ(A_r), but the estimate needed for (4) is a bound on ∑_{X≤q,r≤Y} λ(A_q ∩ A_r). Under the hypothesis ∑_{X≤q≤Y} λ(A_q) ∈ [1,2], the product sum is at most 4, so the proposition is trivial and does not imply (4). The sentence 'Thus we have reduced the proof to showing (4)' is therefore not justified by the proposition as stated. The same error appears in the final display of Section 2.3.1, which concludes ∑_{(q,r)∈[X,Y]^2} λ(A_q)λ(A_r) ≪ 1 rather than the required bound on ∑ λ(A_q ∩ A_r). As printed, the text never derives the variance estimate (4), which is the foundation of the entire reduction. This is a correctable typo, but it is load-bearing.
[Section 2.3.2, Theorem 1 and Propositions 2.4–2.5] Theorem 1, stated in Section 2.3.2 as the 'Main technical result', is not proved in the manuscript, and Proposition 2.3 is proved only conditionally on it. In the proof of Proposition 2.4, the key concentration step after Lemma 2.6 is delegated to [36, Lemma 3.2] and [30, Lemma 2.1], and the proof of Proposition 2.5 invokes Lemmas 2.7 and 2.8 without proof. Reference [36] is listed as an unpublished manuscript ('Proving the Duffin-Schaeffer conjecture without GCD graphs') with no arXiv identifier or publication data, so a reader cannot verify the core combinatorial claim. If the results in [36] are correct, the remaining issues are expository; if they are not, the proof collapses. To support the paper's advertised simplified proof, the author should either include the missing proofs (at least sketches of Theorem 1 and Lemmas 2.7–2.8) or explicitly state that Section 2 is an exposition that assumes the results of [36], which would significantly weaken the claimed contribution.

minor comments (6)

[Section 2, first paragraph] The text says 'the recent proof given by the author with Vazquez and Walker [36]', but reference [36] lists 'M. Hauke, S. Saez, and A. Walker'; the author names should be made consistent.
[Section 2.3.2, Lemma 2.6 and Proposition 2.4] The Hölder conjugate q′ is defined inconsistently: Lemma 2.6 sets q = 2+ε and takes q′ as its conjugate, while Proposition 2.4 defines q′ = 2/(1+2ε). The latter matches the exponent 1/2+ε in (17), so the definition in Lemma 2.6 should be adjusted.
[Section 2.3.1] The notation eψ for a rescaled weight function is confusing, as it may be read as e^ψ; a different symbol such as ψ_j would be clearer.
[Section 1 and Section 2, item 3] There are minor typographical errors: '3th century' should be '3rd century', and 'Koukoulopolus' should be 'Koukoulopoulos'.
[Section 2.1, Theorem 2] The proof of Theorem 2 is omitted; since this result is used in the reduction, please add a reference on where the proof can be found (the current pointer is [7, Theorem 6]).
[Section 2 and Proposition 2.3] The assumptions on ψ are not clearly reconciled: Section 2 restricts to ψ: N→[0,1/2] with square-free support, while Proposition 2.3 allows arbitrary nonnegative ψ on [X,Y]; the paper should state which assumptions are in force for each result.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the Duffin–Schaeffer proof sketch is an exposition of an external theorem, with self-citations to [36] for omitted details but no derivation depending on its own conclusion.

full rationale

The paper does not derive Duffin–Schaeffer from itself. Section 2 is an expository proof sketch of the Koukoulopoulos–Maynard theorem, an external result; the sketch reduces the conjecture to a variance estimate (4), then to a sieve overlap bound (14), then to a combinatorial statement (Prop. 2.3), and finally to Theorem 1. Each reduction is stated with equations; the final theorem's proof is not assumed as input. The main caveats are not circular: Proposition 2.2 is misstated (it bounds the product sum λ(Aq)λ(Ar), which is trivially O(1), instead of the overlap sum λ(Aq∩Ar)), and the final display in §2.3.1 also swaps these sums; this is a typographical/argument gap rather than a self-referential derivation. The key concentration step in Proposition 2.4 is deferred to [36] (an unpublished preprint by the same author) and to the independent [30], and Lemmas 2.7–2.8 are stated without proof with references to [1] and [36]. These are omissions that make the sketch incomplete, but they do not make the conclusion equivalent to an input: the Duffin–Schaeffer theorem is not assumed in order to prove itself, and the cited [36] is a source of the simplification, not a restatement of the target. No fitted parameters or uniqueness theorems from the authors are used. Hence no circularity beyond a minor self-citation for technical details.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The content rests on standard analytic number theory (sieve estimates, Borel-Cantelli, Mertens) and on several unproved technical statements inside the paper. The only nonstandard dependency is the author's unpublished preprint [36], which carries the main technical theorem. There are no empirically fitted parameters and no invented entities.

assumptions (5)

domain assumption Theorem 2 (refined divergence Borel-Cantelli lemma for doubling measures) is used without proof.
Stated in Section 2.1 and used to reduce the Duffin-Schaeffer theorem to a variance estimate; the paper omits its proof and cites [7, Theorem 6].
domain assumption Lemma 2.7 (unweighted anatomy property) is used without proof.
Used in Section 2.3.4 to bound S2; proof omitted with reference to [1, Lemma 10].
domain assumption Lemma 2.8 (divisor anatomy property) is used without proof.
Used in Section 2.3.4 to bound S1; proof omitted with reference to [36, Lemma 4.2].
domain assumption The complementary case psi(q) > 1/2 for infinitely many q is delegated to Pollington-Vaughan [48].
The proof sketch in Section 2 assumes psi(q) <= 1/2; the full theorem relies on [48] for the exceptional case.
domain assumption The proof sketch assumes psi is supported on square-free numbers.
Section 2 states that the general case has 'technical tweaks' not shown; the sketch is not a complete proof for all psi.

how reviews work

0 comments

Cite this review

Pith. "Pith review of A century of metric Diophantine approximation and half a decade since Koukoulopoulos-Maynard." pith.science (2026). https://pith.science/paper/KLBLA4OL

@misc{pith2026250508901,
  author       = {Pith},
  title        = {Pith review of: A century of metric Diophantine approximation and half a decade since Koukoulopoulos-Maynard},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KLBLA4OL}},
  note         = {Machine review of arXiv:2505.08901}
}

read the original abstract

In this note, we review the history of Khintchine's Theorem which is the foundation of metric Diophantine approximation, and discuss several generalizations and recent breakthroughs in this area. We focus particularly on the direction of the Duffin-Schaeffer Conjecture which was spectacularly proven in 2020. We present some simplified key ideas of the proof that can also be applied in various other areas of number theory.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

65 extracted references · 55 canonical work pages

[1]

maliciously chosen

What happens when we replace the Lebesgue measure with some other probability measureµ that is supported on [0, 1)? As long as there is no good reason for this measure to be “maliciously chosen”, the general philosophy is that Khintchine’s Theorem should still hold true with µ in place of the Lebesgue measure. It is obvious that for measures µ that are ab...

work page
[36]

M. Fuchs. On a problem of W. J. LeVeque concerning metric Diophantine approximation. II. Math. Proc. Cambridge Philos. Soc. , 137(1):17–41, 2004

work page 2004
[2]

Closely related to the above is the study of Khintchine’s Theorem on manifolds: Taken a manifold M⊆ Rn that (at least locally) can be described as F (U) where U⊆ Rd is an open set and F is a reasonably smooth map, we use the normalized Lebesgue measure on U to define a probability measure µ onM, and ask about a Khintchine Theorem with respect to µ on the ...

work page
[3]

moving target

Introducing an inhomogeneous parameter: Instead of asking for α− p q ≤ ψ(q) q , we ask for infinitely many solutions to α− p +γ q ≤ ψ(q) q , where γ̸= 0 is an arbitrary fixed real number. The inhomogeneous generalization of Khintchine’s Theorem was proven by Sz¨ usz in 1958 [52], but this is still a very active area of research (for some recent advancemen...

work page 1958
[4]

sufficient independence

Taking the probability-theoretic point of view again, Khintchine’s Theorem is a statement on the measure of the lim sup-set of a sequence of events. If the events were a family of independent events, the statement would follow from a simple application of the Borel–Cantelli Lemma. However, these sets do not satisfy this independence condition, but the ori...

work page
[5]

penalizing

Removing monotonicity: Although having a monotonicity assumption inψ for Khintchine’s Theorem seems natural from the point of view of “penalizing” large denominators, there are, among other reasons, two interesting settings where dealing with a non-monotonic ψ is unavoidable: • Multiplicative Diophantine approximation: Motivated by the famously open conje...

work page 1941
[6]

4This should not take anything away from the original proof

From variance estimate to almost sure : In this part, we will show that the problem can, analogously to Khintchine’s Theorem, be translated into a question on the Lebesgue measure of lim sup q→∞Aq for some particular measurable sets Aq. 4This should not take anything away from the original proof. On the contrary, we believe that it is the duty of the math...

work page 1961
[7]

However, it will turn out to be too much to ask for

The overlap estimate - application of sieve theory : This part deals with upper bounds on λ(Aq∩Ar) λ(Aq)λ(Ar) - if this quantity was bounded by a constant C for all pairs q̸=r, this would immediately show (4). However, it will turn out to be too much to ask for. In this section, most of the work has been developed by Erd˝ os [24] respectively Pollington a...

work page

Show all 65 references

[8]

dyadic splitting

The combinatorial part - the non-existence of a set with only large GCDs : This is the heart of the proof and the key to the breakthrough result of Koukoulopolus–Maynard. It will be stated below as Proposition 2.3 when the corresponding quantities are defined. This part of the...

1930
[9]

smuggle in

(21) To “smuggle in” the additional power of ε in order to show µψ,θ(E′)≤ (µψ(V′)µθ(W′)e−Ct) 1 2 +ε, one only needs to show that µψ(V′)µθ(W′) is not “too small” - in which case even the trivial bound µψ,θ(E′) ⩽µψ(V′)µθ(W′) contradicts assumption (a) of Proposition 2.4. So one ...

2024
[10]

Aistleitner, B

C. Aistleitner, B. Borda, and M. Hauke. On the metric theory of approximations by reduced fractions: A quantitative Koukoulopoulos–Maynard theorem. Compos. Math., 159(2):207–231, 2023

2023
[11]

Aistleitner, T

C. Aistleitner, T. Lachmann, M. Munsch, N. Technau, and A. Zafeiropoulos. The Duffin-Schaeffer conjecture with extra divergence. Adv. Math., 356:106808, 11, 2019

2019
[12]

Alam and A

M. Alam and A. Ghosh. Quantitative rational approximation on spheres. Selecta Math. (N.S.) , 28(5):Paper No. 86, 22, 2022

2022
[13]

B´ enard, W

T. B´ enard, W. He, and H. Zhang. Khintchine dichotomy for self-similar measures. Preprint, arXiv:2409.08061, 2024

2024
[14]

Beresnevich, D

V. Beresnevich, D. Dickinson, and S. Velani. Diophantine approximation on planar curves and the distribution of rational points. Ann. of Math. (2) , 166(2):367–426, 2007. With an Appendix II by R. Vaughan. 22

2007
[15]

Beresnevich, G

V. Beresnevich, G. Harman, A. Haynes, and S. Velani. The Duffin-Schaeffer conjecture with extra divergence II. Math. Z., 275(1-2):127–133, 2013

2013
[16]

Beresnevich, M

V. Beresnevich, M. Hauke, and S. Velani. Borel–Cantelli, zero-one laws and inhomogeneous Duffin– Schaeffer. Preprint, arXiv:2406.19198

arXiv
[17]

Beresnevich and S

V. Beresnevich and S. Velani. A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures. Ann. of Math. (2) , 164(3):971–992, 2006

2006
[18]

Beresnevich and S

V. Beresnevich and S. Velani. Number theory meets wireless communications: an introduction for dummies like us. In Number theory meets wireless communications, Math. Eng., pages 1–67. Springer, 2020

2020
[19]

Beresnevich and S

V. Beresnevich and S. Velani. The divergence Borel–Cantelli lemma revisited. J. Math. Anal. Appl. , 519(1):126750, 2023

2023
[20]

Beresnevich and L

V. Beresnevich and L. Yang. Khintchine’s theorem and Diophantine approximation on manifolds. Acta Math., 231(1):1–30, 2023

2023
[21]

Bj¨ orklund and A

M. Bj¨ orklund and A. Gorodnik. Central limit theorems for Diophantine approximants. Math. Ann., 374(3-4):1371–1437, 2019

2019
[22]

E. Borel. Les probabilit´ es d´ enombrables et leurs applications arithm´ etiques. Rend. Circ. Mat. Palermo, 27:247–271, 1909

1909
[23]

Chow and N

S. Chow and N. Technau. Counting multiplicative approximations. Ramanujan J., 62(1):241–250, 2023

2023
[24]

Chow and N

S. Chow and N. Technau. Littlewood and Duffin-Schaeffer-type problems in Diophantine Approxim- ation. Mem. Amer. Math. Soc. , 296(1475):v+74, 2024

2024
[25]

S. Chow, P. Varju, and H. Yu. Counting rationals and diophantine approximation in missing-digit Cantor sets. Preprint, arXiv:2402.18395, 2024

2024
[26]

Chung and P

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

1952
[27]

Datta and S

S. Datta and S. Jana. On Fourier asymptotics and effective equidistribution. Preprint, arXiv:2407.11961, 2024

2024 arXiv
[28]

Dodson, S

M. Dodson, S. Kristensen, and J. Levesley. A quantitative Khintchine-Groshev type theorem over a field of formal series. Indag. Math. (N.S.) , 16(2):171–177, 2005

2005
[29]

R. J. Duffin and A. C. Schaeffer. Khintchine’s problem in metric Diophantine approximation. Duke Math. J., 8:243–255, 1941

1941
[30]

Einsiedler, A

M. Einsiedler, A. Katok, and E. Lindenstrauss. Invariant measures and the set of exceptions to Littlewood’s conjecture. Ann. of Math. (2) , 164:513–560, 2006. doi:10.4007/annals.2006.164.513

2006 doi
[31]

P. Erd˝ os. On the strong law of large numbers. Trans. Amer. Math. Soc., 67:51–56, 1949

1949
[32]

P. Erd˝ os. Problems and results on diophantine approximations. Compos. Math., 16:52–65, 1964

1964
[33]

P. Erd˝ os. On the distribution of the convergents of almost all real numbers. J. Number Theory , 2:425–441, 1970

1970
[34]

Fr¨ uhwirth and M

L. Fr¨ uhwirth and M. Hauke. The Duffin-Schaeffer Conjecture for multiplicative Diophantine approx- imation. Preprint, arXiv:2403.11257, 2024. 23

2024 arXiv
[35]

M. Fuchs. On a problem of W. J. LeVeque concerning metric Diophantine approximation. Trans. Amer. Math. Soc., 355(5):1787–1801, 2003

2003
[37]

Gallagher

P. Gallagher. Approximation by reduced fractions. J. Math. Soc. Japan , 13:342–345, 1961

1961
[38]

Gallagher

P. Gallagher. Metric simultaneous diophantine approximation. J. Lond. Math. Soc. (2) , s1- 37(1):387–390, 1962

1962
[39]

Green and A

B. Green and A. Walker. Extremal problems for GCDs. Combin. Probab. Comput. , 30(6):922–929, 2021

2021
[40]

Halberstam and H

H. Halberstam and H. Richert. Sieve methods, volume No. 4 of London Mathematical Society Mono- graphs. Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1974

1974
[41]

Hardy and J

G. Hardy and J. Littlewood. Notes on the theory of series. XXIV. A curious power-series. Proc. Cambridge Philos. Soc. , 42:85–90, 1946

1946
[42]

G. Harman. Metric number theory. Clarendon Press, Oxford, England, 1998

1998
[43]

M. Hauke. Quantitative inhomogeneous Diophantine approximation for systems of linear forms. Proc. Amer. Math. Soc., 2023. to appear, preprint: arXiv:2312.01986

2023 arXiv
[44]

Hauke and F

M. Hauke and F. Ram´ ırez. The Duffin-Schaeffer conjecture with a moving target. Preprint, arXiv:2407.05344, 2024

2024 arXiv
[45]

Hauke, S

M. Hauke, S. Saez, and A. Walker. Proving the Duffin-Schaeffer conjecture without GCD graphs
[46]

Khalil and M

O. Khalil and M. Luethi. Random walks, spectral gaps, and Khintchine’s theorem on fractals. Invent. Math., 232(2):713–831, 2023

2023
[47]

Kleinbock and G

D. Kleinbock and G. Margulis. Flows on homogeneous spaces and Diophantine approximation on manifolds. Ann. of Math. (2) , 148(1):339–360, 1998

1998
[48]

Koukoulopoulos, Y

D. Koukoulopoulos, Y. Lamzouri, and J. Lichtman. Erd˝ os’s integer dilation approximation problem and GCD graphs. Preprint, arXiv:2502.09539, 2025

2025 arXiv
[49]

Koukoulopoulos and J

D. Koukoulopoulos and J. Maynard. On the Duffin-Schaeffer conjecture. Ann. of Math. (2) , 192(1):251–307, 2020

2020
[50]

Koukoulopoulos, J

D. Koukoulopoulos, J. Maynard, and D. Yang. An almost sharp quantitative version of the Duffin- Schaeffer conjecture. Duke Math. J. , 2024. to appear, preprint: arXiv:2404.14628

2024 arXiv
[51]

W. LeVeque. On the frequency of small fractional parts in certain real sequences. Trans. Amer. Math. Soc., 87:237–261, 1958

1958
[52]

W. J. LeVeque. On the frequency of small fractional parts in certain real sequences. II. Trans. Amer. Math. Soc., 94:130–149, 1960

1960
[53]

K. Mahler. Some suggestions for further research. Bull. Austral. Math. Soc. , 29(1):101–108, 1984

1984
[54]

Paley and A

R. Paley and A. Zygmund. On some series of functions. I, II. Proc. Camb. Philos. Soc., 26:337–357, 458–474, 1930

1930
[55]

Paley and A

R. Paley and A. Zygmund. On some series of functions. III. Proc. Camb. Philos. Soc. , 28:190–205, 1932

1932
[56]

Pfitscher

R. Pfitscher. Counting rational approximations on rank one flag varieties. Preprint, arXiv:2410.16508, 24 2024

2024 arXiv
[57]

Pollington and R

A. Pollington and R. Vaughan. The k-dimensional Duffin and Schaeffer conjecture. Mathematika, 37(2):190–200, 1990. doi:10.1112/S0025579300012900

1990 doi
[58]

Schindler, R

D. Schindler, R. Srivastava, and N. Technau. Rational Points Near Manifolds, Homogeneous Dy- namics, and Oscillatory Integrals. Preprint, arXiv:2310.03867, 2023

2023 arXiv
[59]

W. Schmidt. A metrical theorem in diophantine approximation. Canad. J. Math., 12, 619–631 , 1960

1960
[60]

W. Schmidt. Metrical theorems on fractional parts of sequences. Trans. Amer. Math. Soc., 110:493– 518, 1964

1964
[61]

P. Sz¨ usz. ¨ uber die metrische Theorie der Diophantischen Approximation. Acta Math. Acad. Sci. Hungar., 9:177–193, 1958

1958
[62]

Vaughan and S

R. Vaughan and S. Velani. Diophantine approximation on planar curves: the convergence theory. Invent. Math., 166(1):103–124, 2006

2006
[63]

H. Weyl. ¨Uber die Gleichverteilung von Zahlen mod. Eins. Math. Ann., 77(3):313–352, 1916

1916
[64]

H. Yu. On the metric theory of inhomogeneous Diophantine approximation: an Erd˝ os-Vaaler type result. J. Number Theory, 224:243–273, 2021

2021
[65]

H. Yu. Rational points near self-similar sets. Preprint, arXiv:2101.05910, 2021. 25

2021 arXiv

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

maliciously chosen

What happens when we replace the Lebesgue measure with some other probability measureµ that is supported on [0, 1)? As long as there is no good reason for this measure to be “maliciously chosen”, the general philosophy is that Khintchine’s Theorem should still hold true with µ in place of the Lebesgue measure. It is obvious that for measures µ that are ab...

work page

[2] [36]

M. Fuchs. On a problem of W. J. LeVeque concerning metric Diophantine approximation. II. Math. Proc. Cambridge Philos. Soc. , 137(1):17–41, 2004

work page 2004

[3] [2]

Closely related to the above is the study of Khintchine’s Theorem on manifolds: Taken a manifold M⊆ Rn that (at least locally) can be described as F (U) where U⊆ Rd is an open set and F is a reasonably smooth map, we use the normalized Lebesgue measure on U to define a probability measure µ onM, and ask about a Khintchine Theorem with respect to µ on the ...

work page

[4] [3]

moving target

Introducing an inhomogeneous parameter: Instead of asking for α− p q ≤ ψ(q) q , we ask for infinitely many solutions to α− p +γ q ≤ ψ(q) q , where γ̸= 0 is an arbitrary fixed real number. The inhomogeneous generalization of Khintchine’s Theorem was proven by Sz¨ usz in 1958 [52], but this is still a very active area of research (for some recent advancemen...

work page 1958

[5] [4]

sufficient independence

Taking the probability-theoretic point of view again, Khintchine’s Theorem is a statement on the measure of the lim sup-set of a sequence of events. If the events were a family of independent events, the statement would follow from a simple application of the Borel–Cantelli Lemma. However, these sets do not satisfy this independence condition, but the ori...

work page

[6] [5]

penalizing

Removing monotonicity: Although having a monotonicity assumption inψ for Khintchine’s Theorem seems natural from the point of view of “penalizing” large denominators, there are, among other reasons, two interesting settings where dealing with a non-monotonic ψ is unavoidable: • Multiplicative Diophantine approximation: Motivated by the famously open conje...

work page 1941

[7] [6]

4This should not take anything away from the original proof

From variance estimate to almost sure : In this part, we will show that the problem can, analogously to Khintchine’s Theorem, be translated into a question on the Lebesgue measure of lim sup q→∞Aq for some particular measurable sets Aq. 4This should not take anything away from the original proof. On the contrary, we believe that it is the duty of the math...

work page 1961

[8] [7]

However, it will turn out to be too much to ask for

The overlap estimate - application of sieve theory : This part deals with upper bounds on λ(Aq∩Ar) λ(Aq)λ(Ar) - if this quantity was bounded by a constant C for all pairs q̸=r, this would immediately show (4). However, it will turn out to be too much to ask for. In this section, most of the work has been developed by Erd˝ os [24] respectively Pollington a...

work page