Toward Khintchine's theorem with a moving target: extra divergence or finitely centered target

Felipe A. Ram\'irez; Gilbert Michaud

REVIEW 2 major objections 3 minor 1 cited by

Toward Khintchine's theorem with a moving target: extra divergence or finitely centered target

T0 review · 2 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The moving-target conjecture for Khintchine's theorem is proved in two regimes: under an extra-divergence condition on the approximation function, and when the moving centers lie in a finite set.

desk verdict A genuinely new partial proof of the moving-target Khintchine conjecture, with a fixable threshold typo in the finite-center argument that a referee should catch. read the letter →

arxiv 2506.04187 v2 pith:HJBLPT5K submitted 2025-06-04 math.NT

Gilbert Michaud , Felipe A. Ramírez This is my paper

classification math.NT MSC 11J8311K60

keywords Diophantineapproximationinhomogeneousshrinkingtargetsmovinglimsupsetsquasi-independenceonaveragedivisorfunctionKhintchine'stheorem

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 concerns the classical inhomogeneous Khintchine theorem, which says that for a fixed offset, almost every real number is approximated infinitely often by shifted rationals with error governed by a decreasing function, provided its sum diverges. It addresses the open case where the offset is allowed to depend on the denominator, so the target both shrinks and moves. The paper proves the moving-target conjecture for two classes: approximation functions satisfying an extra-divergence condition, effectively requiring a divergent sum after dividing by a square-root-of-log factor and iterated logarithms, and target-center sequences contained in a finite set. In both cases the set of approximated points has full Lebesgue measure. A byproduct is a finite-coloring theorem: in any finite coloring of the denominators, one color class alone supports full-measure approximation.

What carries the argument

The central object is the limsup set $W(\psi,\gamma)=\{\alpha\in[0,1]:\|q\alpha-\gamma_q\|<\psi(q)\text{ infinitely often}\}$, encoding visits of the orbit $q\alpha\bmod 1$ to shrinking intervals centered at $\gamma_q$. Theorem 1 runs on an overlap estimate $m(A_q\cap A_r)\le 2m(A_q)m(A_r)+\frac{\gcd(q,r)}{q}m(A_q)$ and an abstract quasi-independence lemma that absorbs the gcd term under an extra-divergence condition. A divisor-count transfer converts the assumed divergence into divergence of $\sum \psi(q)/(f(\log d(q))d(q))$, using the central limit theorem for the divisor function and the estimate $\sum_{n\le x}1/d(n)\asymp x/\sqrt{\log x}$. Theorem 3 instead uses a fixed-center quasi-independence estimate and a pigeonhole step that selects one color class, together with an equidistribution statement for the sets of admissible shifted numerators; the full-measure conclusion then follows from a Borel–Cantelli-type full-measure criterion.

What would settle it

A concrete check: fix $\psi(q)=1/q$ and let $\gamma_q$ be the fractional part of $\sqrt{q}$. The theorem requires $m(W(\psi,\gamma))=1$. Compute the intersection sum $\sum_{q,r\le Q}m(A_q\cap A_r)$ and the quasi-independence ratio used in the overlap argument; if the ratio does not stay bounded away from zero as $Q\to\infty$, the overlap absorption has failed for this target. Alternatively, a single explicit sequence $(\gamma_q)$ and decreasing $\psi$ with divergent extra-divergence sum but $m(W(\psi,\gamma))<1$ would refute the theorem outright.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the moving-target conjecture holds under two additional hypotheses. The first, Theorem 1, states that if for some $\varepsilon>0$ and $k\ge 2$ the sum $\sum_{q=1}^{\infty} \psi(q)/(\sqrt{\log q}\,(\log\log q)\cdots(\log^{(k)} q)^{1+\varepsilon})$ diverges, then $m(W(\psi,\gamma))=1$ for every sequence $(\gamma_q)$ of centers. The second, Theorem 3 and Corollary 4, requires only the classical divergence $\sum\psi(q)=\infty$ but constrains the centers to a finite set $\{\sigma_1,\dots,\sigma_\ell\}$; then there is some $k$ such that the restricted limsup set $W(1_{\{\gamma_q=\sigma_k\}}\psi,\sigma_k)$ has full measure. Theorem 3 also yields a finite-colorings statement: for any finite partition of $\mathbb{N}$, some cell $\pi$ satisfies $m(W(1_\pi\psi,\gamma))=1$ for any fixed $\gamma$. These are genuine extensions of the fixed-center theorem, not just convergence-side observations.

Load-bearing premise

The entire argument rests on being able to absorb the overlap term $\gcd(q,r)\psi(q)/q$ into the square of the sum of the measures; if that absorption fails at the stated divergence thresholds, the quasi-independence argument yields only a positive-measure set rather than full measure.

Editorial extensions

If this is right

For any decreasing $\psi$ satisfying the extra-divergence condition, $m(W(\psi,\gamma))=1$ for every moving target; in particular this covers $\psi(q)=1/q$.
If the centers $\gamma_q$ are drawn from a finite set, the classical divergence condition $\sum\psi(q)=\infty$ alone is enough, and one of the finitely many centers is responsible for full measure on its own subsequence.
In any finite coloring of the denominators, some color class $\pi$ yields $m(W(1_\pi\psi,\gamma))=1$, giving rational approximations with monochromatic denominators.
A fast-divergence asymptotic formula previously covered extremely large error sums; Theorem 1 lowers the threshold to a tractable extra-divergence condition that covers natural functions like $1/q$.

Reading between the lines

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

The extra $\sqrt{\log q}$ factor in the threshold is likely an artifact of the inhomogeneous gcd term, and a sharper overlap estimate could plausibly lower it to $\sum\psi(q)(\log q)^{-\varepsilon}=\infty$, matching the analogous extra-divergence results the paper cites.
The finite-color theorem suggests a route to the full conjecture: by the paper's own reduction it suffices to prove the conjecture for centers in a countable dense set such as the rationals, so a pigeonhole or quasi-independence argument over a countable family of centers might close the problem.
The divisor-count transfer indicates the overlap term is essentially controlled by the divisor function; testing whether $\sum \psi(q)/d(q)=\infty$ is already sufficient would be a direct stress test of the method.

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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

Two things to know. First, this is the first real progress on Hauke–Ramirez's moving-target Khintchine conjecture: they prove it under an extra-divergence condition and for finitely many centers, plus a monochromatic-denominator corollary. Second, there is a real gap in Proposition 18 as written, but it looks like a typo, not a fatal flaw.

The extra-divergence theorem is new and nontrivial: it covers psi(q)=1/q, which Sprindzuk's fast-divergence asymptotic cannot, and the proof via Yu's abstraction plus Kac's divisor CLT is clean. The finite-center theorem is also new, and the pigeonhole reduction to Schmidt's fixed-center QIA is a neat idea. The paper is honest about the literature: the conjecture from [17] is the target of the proof, not an input, and the citation pattern is not circular.

The soft spot is Proposition 18. The intersection condition for A'_q and A'_r is |(a+gamma)r - (b+gamma)q| < r psi(q) + q psi(r), but M(q,r) is defined with the smaller threshold 2r psi(q). Since psi is decreasing, r psi(q) <= q psi(r), so the true threshold is between q psi(r) and 2q psi(r). As written, the bound m(A'_q cap A'_r) <= (2 psi(q)/q) M(q,r) is false: M undercounts the intersecting pairs. The subsequent estimates are consistent with replacing 2r psi(q) by 2q psi(r); then the first term gives M << q psi(r) when q psi(r) >= gcd(q,r), and the third term works because r psi(q) <= q psi(r) lets Lemma 17 apply with delta=1/5. So the fix is straightforward and Theorem 3 should go through, but the proof is incomplete until that correction is made and verified. This is exactly the kind of thing a referee should catch and the authors should fix.

The paper ships no code; it relies on standard black boxes (Schmidt, Yu, Beresnevich–Hauke–Velani), which is normal for this area. The extra-divergence threshold with iterated logs is natural and the exposition is clear.

Who is it for: people in metric Diophantine approximation and shrinking-target dynamics. It deserves a serious referee. I would send it to review, with instructions to have the authors correct the threshold in Proposition 18 and reverify the three-term split.

Referee Report

2 major / 3 minor

Summary. This paper studies the one-dimensional inhomogeneous Khintchine theorem with a moving target. Defining W(ψ,γ) = {α∈[0,1]: ||qα−γ_q||<ψ(q) infinitely often}, the authors prove two partial results toward the Hauke–Ramírez Conjecture 1: Theorem 1 gives full measure under an extra-divergence condition, and Theorem 3 gives full measure when the centers γ_q lie in a finite set, with Corollary 5 on monochromatic denominators. The proof of Theorem 1 combines an overlap estimate (Lemma 15), a quasi-independence criterion (Proposition 8), and divisor-sum estimates via Kac's theorem (Lemma 11). The proof of Theorem 3 uses Schmidt's fixed-center QIA estimate (Proposition 18) and a pigeonhole argument to pass to one color class, then applies a Borel–Cantelli criterion from Beresnevich–Hauke–Velani.

Significance. If the main theorems are corrected as indicated below, the paper would constitute a genuine advance on a conjecture from Hauke and Ramírez, and Corollary 5 is a new monochromatic-denominator inhomogeneous Khintchine theorem. The paper is written carefully, uses appropriate external benchmarks (Szüsz, Schmidt, Sprindžuk, BDV, BHV, Kac), and its abstract QIA machinery is a useful framework. The main proofs are not machine-checked, but they are detailed enough to review line by line. Two load-bearing points currently need repair: the printed extra-divergence hypothesis of Theorem 1 does not match the proof, and the QIA estimate in Proposition 18 uses an incorrect intersection threshold. Both appear fixable within the manuscript's framework.

major comments (2)

[§1.1 / §4.1 (Theorem 1)] The hypothesis of Theorem 1 is printed as ∑ ψ(q)√(log q (log log q)...(log...log (k iterates) q)^{1+ε}) = ∞. Since the radical is at least 1 for all sufficiently large q, this condition is implied by the plain divergence ∑ψ(q)=∞; it is therefore not an 'extra divergence' assumption and, as stated, the theorem would claim the full Conjecture 1. The proof in §4.1 instead uses the condition ∑ψ(q)/(f(log log q)√log q)=∞ with f(x)=x(log x)...(log...log (k−2 iterates) x)^{1+ε}, i.e. the square root appears in the denominator. These are different hypotheses. The statement in the abstract, the introduction, and the proof must be reconciled (most likely by placing the radical in the denominator), and Corollary 2 and the 'in particular' remark should be checked against the corrected condition.
[§5, Proposition 18] The proof of Proposition 18 defines M(q,r) with the threshold |(a+γ)r−(b+γ)q| < 2rψ(q) and then asserts m(A'_q∩A'_r) ≤ (2ψ(q)/q)M(q,r). The actual nonempty-intersection threshold is rψ(q)+qψ(r). The asserted inequality is false in general: for γ=0, ψ(n)=1/n, q=100, r=10, one has rψ(q)+qψ(r)=10.1 and, e.g., the pair a=9, b=1 gives an intersection, while 2rψ(q)=0.2 and M(100,10)=0 for the reduced sets S(100), S(10). Thus the QIA estimate ∑_{q,r≤Q} m(A'_q∩A'_r) ≪ Ψ(Q)², which is the load-bearing input for Theorem 3, is not proved as written. The fix is to replace the threshold by rψ(q)+qψ(r) (or by a comparable upper bound such as 2qψ(r)); the subsequent estimates in the proof appear consistent with this correction because rψ(q) ≤ qψ(r), but the proposition should be re-verified with the corrected threshold.

minor comments (3)

[§6, proof of Theorem 3] In the verification of condition (8), the displayed chain m(A'_{q,k}∩I) ≤ (ψ(q)/q)#(S(q)/q∩I) misses the factor 2 coming from the length of each component interval; as written it is false, e.g. when I is one of the component intervals of A'_{q,k}. Replacing ψ(q)/q by 2ψ(q)/q repairs the chain and still gives (8).
[§4.1] In the application of Proposition 8, η(q,r)=gcd(q,r)/q can be smaller than 1, while Proposition 8 states η:N²→[1,∞). This can be fixed by replacing η with max{η,1} or by extending the statement to nonnegative η, but the mismatch should be addressed.
[§1.1 and §4.1] The iterated-logarithm notation in Theorem 1, Corollary 2, and the proof is ambiguous; please define the number of iterates explicitly, for example with log_k x, so that the exponents in the statement and in the definition of f(x) can be checked.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the derivation is self-contained against external benchmarks.

full rationale

The central claim (Conjecture 1 from [17]) is the target of the proof, not an input. The paper proves the moving-target conjecture under extra divergence and for finitely centered targets, and nowhere does it fit a parameter to a subset of the target quantity and then rename that fit as a prediction. Theorem 1 is built from standard quasi-independence machinery: Proposition 8 follows Yu's argument, Lemma 11 uses Kac's central limit theorem and the classical estimate sum_{n<=x} 1/d(n) asy x/sqrt(log x), Lemma 15 is an overlap bound, and Lemma 16 is a local-density input; none of these inputs contains the conclusion. Theorem 3 imports Schmidt's fixed-center quasi-independence estimate (Proposition 18) and verifies the hypotheses of the external Beresnevich–Hauke–Velani Proposition 10; no step in that chain reduces to the target statement by definition. The paper's self-citations, such as [17] for the conjecture and [4] and [22] for context or technical extensions, are motivational or auxiliary rather than load-bearing, and the cited external results (Szüsz, Schmidt, Sprindžuk, BDV, Kac, Tenenbaum) supply independent support. The skeptical concern about the threshold in Proposition 18 is a potential correctness/local-error issue in an estimate, not an instance of circularity: a wrong intermediate bound would make the proof incomplete, but it would not make the claimed theorem equivalent to its own hypotheses.

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

The paper introduces no new entities and no fitted parameters. Its results rest on classical theorems in Diophantine approximation and analytic number theory, cited and used as black boxes; the non-black-box ingredients are elementary lemmas proved in the text.

assumptions (5)

standard math Sprindzuk's asymptotic formula [24, Theorem 18] for the counting function N(Q,alpha) in inhomogeneous Diophantine approximation.
Used in Proposition 14 to show that fast divergence implies the conjecture; the main new theorems do not rely on this proposition, but it is stated and applied.
standard math Schmidt's fixed-center quasi-independence estimate [23, Proposition 2], reproduced as Proposition 18 via Lemma 17.
Basis for the finite-centered Theorem 3; Lemma 17 is proved in the text but the overall QIA framework is inherited from Schmidt.
standard math Kac's central limit theorem for log_2 d(n) [19] and the estimate sum_{n <= x} 1/d(n) asymp x/sqrt(log x) [26, Theorem II.6.8].
Used in Lemma 11 to transfer the extra-divergence weight to divisor sums, a load-bearing step for Theorem 1.
standard math Beresnevich-Hauke-Velani zero-one criterion [7, Theorem 5].
Used in Theorem 3 to upgrade positive measure on every ball to full measure; its hypotheses are verified in the proof.
standard math Divergence Borel-Cantelli and Chung-Erdos lemmas [13, 11].
Core measure-theoretic engine behind Proposition 8 and the overall quasi-independence method.

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Cite this review

Pith. "Pith review of Toward Khintchine's theorem with a moving target: extra divergence or finitely centered target." pith.science (2026). https://pith.science/paper/HJBLPT5K

@misc{pith2026250604187,
  author       = {Pith},
  title        = {Pith review of: Toward Khintchine's theorem with a moving target: extra divergence or finitely centered target},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HJBLPT5K}},
  note         = {Machine review of arXiv:2506.04187}
}

abstract

Sz{\"u}sz's inhomogeneous version (1958) of Khintchine's theorem (1924) gives conditions on $\psi:\mathbb{N}\to\mathbb{R}_{\geq 0}$ under which for almost every real number $\alpha$ there exist infinitely many rationals $p/q$ such that \begin{equation*} \lvert\alpha - \frac{p+\gamma}{q}\rvert < \frac{\psi(q)}{q}, \end{equation*} where $\gamma\in\mathbb{R}$ is some fixed inhomogeneous parameter. It is often interpreted as a statement about visits of $q\alpha\,(\bmod 1)$ to a shrinking target centered around $\gamma\,(\bmod 1)$, viewed in $\mathbb{R}/\mathbb{Z}$. Hauke and the second author have conjectured that Sz{\"u}sz's result continues to hold if the target is allowed to move as well as shrink, that is, if the inhomogeneous parameter $\gamma$ is allowed to depend on the denominator $q$ of the approximating rational. We show that the conjecture holds under an ``extra divergence'' assumption on $\psi$. We also show that it holds when the inhomogeneous parameter's movement is constrained to a finite set. As a byproduct, we obtain a finite-colorings version of the inhomogeneous Khintchine theorem, giving rational approximations with monochromatic denominators.

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [17]

Hauke and F

M. Hauke and F. A. Ramirez. The Duffin–Schaeffer conjecture with a moving target, 2024. 2, 3, 5 22 GILBERT MICHAUD AND FELIPE A. RAMÍREZ

work page 2024

[2] [1]

A note on the Duffin–Schaeffer conjecture with slow divergence.Bulletin of the London Mathematical Society, 46(1):164–168, 2014

Christoph Aistleitner. A note on the Duffin–Schaeffer conjecture with slow divergence.Bulletin of the London Mathematical Society, 46(1):164–168, 2014. 5

work page 2014

[3] [2]

The Duffin-Schaeffer conjecture with extra divergence.Adv

Christoph Aistleitner, Thomas Lachmann, Marc Munsch, Niclas Technau, and Agamemnon Zafeiropou- los. The Duffin-Schaeffer conjecture with extra divergence.Adv. Math., 356:106808, 11, 2019. 3

work page 2019

[4] [3]

Allen and F

D. Allen and F. A. Ramirez. Inhomogeneous approximation for systems of linear forms with primitivity constraints.Math. Z., to appear. 2

work page

[5] [4]

Allen and F

D. Allen and F. A. Ramírez. Independence inheritance and Diophantine approximation for systems of linear forms.Int. Math. Res. Not. IMRN, (2):1760–1794, 2023. 2

work page 2023

[6] [5]

Beresnevich, D

V. Beresnevich, D. Dickinson, and S. Velani. Measure theoretic laws for lim sup sets.Mem. Amer. Math. Soc., 179(846):x+91, 2006. 8

work page 2006

[7] [6]

The Duffin-Schaeffer conjecture with extra divergence II.Math

Victor Beresnevich, Glyn Harman, Alan Haynes, and Sanju Velani. The Duffin-Schaeffer conjecture with extra divergence II.Math. Z., 275(1-2):127–133, 2013. 3, 5

work page 2013

[8] [7]

Borel-cantelli, zero-one laws and inhomogeneous duffin-schaeffer.arXiv preprint arXiv:2406.19198, 2024

Victor Beresnevich, Manuel Hauke, and Sanju Velani. Borel-cantelli, zero-one laws and inhomogeneous duffin-schaeffer.arXiv preprint arXiv:2406.19198, 2024. 8, 19

arXiv 2024