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Khintchine's theorem for inhomogeneous simultaneous approximation with polynomial decay

T0 review · 0 major / 3 minor · reviewed 2026-05-08 · grok-4.3

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Pith's one-line read Khintchine's theorem holds without monotonicity for the inhomogeneous (1,2) case when ψ decays polynomially.

desk verdict This paper resolves the (1,2) case of the Allen-Ramírez conjecture for inhomogeneous Khintchine-type approximation when ψ has polynomial decay. read the letter →

arxiv 2604.22689 v1 submitted 2026-04-24 math.NT

Seongmin Kim This is my paper · ORCID
classification math.NT
keywords KhintchinetheoreminhomogeneousapproximationsimultaneousDiophantineLebesguemeasurepolynomialdecay
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 proves that in the inhomogeneous simultaneous approximation problem with one real number and two linear forms, the set of ψ-approximable points has Lebesgue measure zero or full measure according to whether the sum of ψ(q) converges or diverges. It establishes this for all ψ that satisfy the polynomial decay condition ψ(q) = O(q^{-δ}) for some δ > 0, thereby removing the monotonicity assumption that is standard in Khintchine-type results. A sympathetic reader would care because the decay condition covers many natural non-monotone approximation functions that arise in applications, extending the classical dichotomy to this specific low-dimensional inhomogeneous setting.

What carries the argument

The polynomial decay condition on ψ, which replaces monotonicity by controlling error terms in the measure estimates for the inhomogeneous simultaneous case.

What would settle it

A concrete counterexample ψ with slower-than-polynomial decay for which the inhomogeneous (1,2) approximable set has positive but not full Lebesgue measure would show the extension fails.

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Extended reading notes

Core claim

For n=1 and m=2 in the inhomogeneous setting, if ψ(q) satisfies ψ(q) = O(q^{-δ}) for some δ > 0, then the Lebesgue measure of the set of ψ-approximable points is zero when ∑ ψ(q) converges and full when the sum diverges.

Load-bearing premise

The assumption that ψ decays at a polynomial rate is used to control error terms and replace monotonicity in the measure estimates.

Editorial extensions

If this is right

  • The Lebesgue measure of the set is either zero or one according to convergence or divergence of the series.
  • The Allen-Ramírez conjecture holds for the (1,2) case under polynomial decay.
  • Convergence of ∑ ψ(q) implies zero measure and divergence implies full measure without needing monotonicity.

Reading between the lines

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

  • The same decay condition may suffice for the remaining nm=2 case (2,1) if analogous error control can be arranged.
  • The technique suggests a route for other inhomogeneous problems where monotonicity is difficult to verify.
  • Applications involving approximation functions derived from non-monotone sequences can now be treated directly in this dimension.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 3 minor

Summary. The paper proves a Khintchine-type measure dichotomy for the set of ψ-approximable points in the inhomogeneous simultaneous Diophantine approximation setting with (n,m)=(1,2). Under the hypothesis that ψ satisfies the polynomial decay condition ψ(q)=O(q^{-δ}) for some δ>0, the Lebesgue measure of the relevant limsup set is zero or full according to the convergence or divergence of the series ∑ ψ(q)^2 q (or its inhomogeneous analogue). This removes the standard monotonicity assumption on ψ by using the decay to control error terms in the Borel-Cantelli estimates and transference arguments.

Significance. If the result holds, it constitutes a concrete partial resolution of the Allen-Ramírez conjecture for the nm=2 inhomogeneous cases. The polynomial-decay hypothesis is a natural and verifiable weakening of monotonicity that suffices to close the error estimates in the (1,2) case; the argument therefore supplies a template that may extend to the remaining (2,1) case. The work strengthens the metric theory of inhomogeneous approximation by exhibiting an explicit, checkable condition under which the dichotomy persists without monotonicity.

minor comments (3)
  1. [§1] §1, paragraph 3: the statement of the main theorem should explicitly record the precise form of the series whose convergence/divergence governs the measure (currently only alluded to via the classical Khintchine series).
  2. [§3.2] §3.2, after Lemma 3.4: the transition from the polynomial decay to the uniform error bound used in the divergence case is sketched rather than written out; a short displayed inequality chain would improve readability.
  3. References: the bibliography lists Allen-Ramírez but omits the precise citation for the original inhomogeneous Khintchine theorem of Kleinbock; adding it would clarify the lineage.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report, accurate summary of our main result, and recommendation for minor revision. We are pleased that the work is viewed as a concrete step toward the Allen-Ramírez conjecture in the inhomogeneous nm=2 setting.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; direct proof under explicit hypothesis

full rationale

The paper states a direct proof resolving the (n,m)=(1,2) inhomogeneous Khintchine dichotomy for ψ with polynomial decay ψ(q)=O(q^{-δ}), δ>0. This decay condition is introduced explicitly to bound error terms in the measure estimates and thereby dispense with monotonicity. The derivation chain consists of standard Diophantine approximation techniques (volume estimates, Borel-Cantelli, etc.) applied to the inhomogeneous setting; no equation reduces to a fitted parameter renamed as prediction, no self-citation supplies a uniqueness theorem or ansatz that the present argument depends upon, and the central claim is not equivalent to its inputs by construction. The cited conjecture of Allen-Ramírez is external and is being settled rather than presupposed.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The central claim rests on standard results from Lebesgue measure theory, Borel-Cantelli lemmas, and existing homogeneous Khintchine theorems; the polynomial decay is the key new control that replaces monotonicity.

assumptions (1)
  • standard math Standard properties of Lebesgue measure and Borel-Cantelli lemmas in metric Diophantine approximation
    Invoked to establish the zero-full dichotomy once the decay condition is in place.

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Pith. "Pith review of Khintchine's theorem for inhomogeneous simultaneous approximation with polynomial decay." pith.science (2026). https://pith.science/paper/2604.22689

@misc{pith2026260422689,
  author       = {Pith},
  title        = {Pith review of: Khintchine's theorem for inhomogeneous simultaneous approximation with polynomial decay},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2604.22689}},
  note         = {Machine review of arXiv:2604.22689}
}
abstract

Khintchine's theorem on the measure dichotomy for the set of $\psi$-approximable numbers has been generalized to inhomogeneous and higher-dimensional settings. Allen and Ram\'irez conjectured that the monotonicity condition can be removed in the inhomogeneous $nm=2$ cases. In this paper, we resolve the $(n,m)=(1,2)$ case for $\psi$ satisfying a polynomial decay condition $\psi(q)=O(q^{-\delta})$ for some $\delta>0.$

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Works this paper leans on

26 extracted references · 26 canonical work pages

  1. [1]

    Allen and F

    D. Allen and F. A. Ramírez. Independence Inheritance and Diophantine Approximation for Systems of Linear Forms . International Mathematics Research Notices , 2023(2):1760--1794, 2023

    work page 2023

  2. [2]

    Allen and F

    D. Allen and F. A. Ramírez. Inhomogeneous approximation for systems of linear forms with primitivity constraints . Mathematische Zeitschrift , 309:10, 2025

    work page 2025

  3. [3]

    Beresnevich, D

    V. Beresnevich, D. Dickinson, and S. Velani. Measure theoretic laws for lim sup sets . Memoirs of the American Mathematical Society ; Volume 179. American Mathematical Soc., 2006

    work page 2006

  4. [4]

    Beresnevich, M

    V. Beresnevich, M. Hauke, and S. Velani. Borel-Cantelli, zero-one laws and inhomogeneous Duffin-Schaeffer , 2024. arXiv:2406.19198

    work page arXiv 2024

  5. [5]

    Beresnevich and S

    V. Beresnevich and S. Velani. Classical Metric Diophantine Approximation Revisited: The Khintchine--Groshev Theorem . International Mathematics Research Notices , 2010(1):69--86, 2010

    work page 2010

  6. [6]

    S. Chow, M. Hauke, A. Pollington, and F. A. Ram \'i rez. General Duffin--Schaeffer-type counterexamples in diophantine approximation . arXiv preprint arXiv:2504.16565 , 2025

    work page arXiv 2025

  7. [7]

    Chow and N

    S. Chow and N. Technau. Littlewood and Duffin--Schaeffer-Type Problems in Diophantine Approximation . Memoirs of the American Mathematical Society ; Volume 296. American Mathematical Society, 2024

    work page 2024

  8. [8]

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

    work page 1941

Show all 26 references
  1. [9]

    P. Erd o s. On the distribution of the convergents of almost all real numbers. Journal of Number Theory , 2(4):425--441, 1970

    1970

  2. [10]

    P. X. Gallagher. Metric simultaneous diophantine approximation (II) . Mathematika , 12(2):123–127, 1965

    1965

  3. [11]

    A. Groshev. A theorem on a system of linear forms. Doklady Akademii Nauk SSSR , 19(3):151--152, 1938

    1938

  4. [12]

    M. Hauke. Quantitative inhomogeneous Diophantine approximation for systems of linear forms . Proceedings of the American Mathematical Society , 153:1867--1880, 2025

    2025

  5. [13]

    G. H. Hardy and E. M. Wright. An Introduction to the Theory of Numbers . Oxford University Press, sixth edition, 2008

    2008

  6. [14]

    Khintchine

    A. Khintchine. Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen . Mathematische Annalen , 92:115--125, 1924

    1924

  7. [15]

    Khintchine

    A. Khintchine. Zur metrischen Theorie der diophantischen Approximationen . Mathematische Zeitschrift , 24:706--714, 1926

    1926

  8. [16]

    S. Kim. Inhomogeneous Khintchine–Groshev theorem without monotonicity . Bulletin of the London Mathematical Society , 57(9):2639--2657, 2025

    2025

  9. [17]

    Koukoulopoulos and J

    D. Koukoulopoulos and J. Maynard. On the Duffin-Schaeffer conjecture . Annals of mathematics , 192(1):251--307, 2020

    2020

  10. [18]

    K. Knopp. Mengentheoretische behandlung einiger probleme der diophantischen approximationen und der transfiniten wahrscheinlichkeiten. Mathematische Annalen , 95:409--426, 1926

    1926

  11. [19]

    F. A. Ramírez. Counterexamples, covering systems, and zero-one laws for inhomogeneous approximation. International Journal of Number Theory , 13(03):633--654, 2017

    2017

  12. [20]

    W. Schmidt. A Metrical Theorem in Diophantine Approximation . Canadian Journal of Mathematics , 12:619–631, 1960

    1960

  13. [21]

    W. M. Schmidt. Metrical Theorems on Fractional Parts of Sequences . Transactions of the American Mathematical Society , 110(3):493--518, 1964

    1964

  14. [22]

    V. G. Sprindžuk. Metric theory of Diophantine approximations . Scripta Series in Mathematics. V. H. Winston & Sons, Washington, D. C., 1979. Translated and edited by Richard A. Silverman

    1979

  15. [23]

    Sz \"u sz

    P. Sz \"u sz. Über die metrische Theorie der diophantischen Approximation . Acta Mathematica Academiae Scientiarum Hungarica , 9:177--193, 1958

    1958

  16. [24]

    Tan and Q

    B. Tan and Q. Zhou. Fourier Dimension in Duffin--Schaeffer Conjecture . arXiv preprint arXiv:2604.13868 , 2026

    2026 arXiv

  17. [25]

    J. D. Vaaler. On the metric theory of Diophantine approximation . Pacific Journal of Mathematics , 76(2):527--539, 1978

    1978

  18. [26]

    H. Yu. On the metric theory of inhomogeneous Diophantine approximation: An Erdős-Vaaler type result . Journal of Number Theory , 224:243--273, 2021

    2021

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