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arxiv: 2605.19582 · v1 · pith:LZKO5BLXnew · submitted 2026-05-19 · 🧮 math.NT

The inhomogeneous Khintchine Theorem in dimension two

Demi Allen , Manuel Hauke-Treuer , Felipe A. Ram\'irez This is my paper

Pith reviewed 2026-05-20 02:15 UTC · model grok-4.3

classification 🧮 math.NT
keywords inhomogeneous Diophantine approximationKhintchine theoremmetric number theorydimension twomonotonicity assumptionGroshev theoremDani correspondence
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The pith

The inhomogeneous Khintchine theorem holds in dimension two without a monotonicity assumption.

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

The paper proves that the inhomogeneous variant of Khintchine's theorem is valid in two dimensions even when the approximating function is not monotonic. This completes the metric theory of inhomogeneous Diophantine approximation across all dimensions, matching the homogeneous case where monotonicity is also unnecessary. A reader would care because it resolves the last open case and confirms a conjecture about systems of linear forms. The result shows that the inhomogeneous theory now aligns fully with its homogeneous counterpart.

Core claim

We prove that the inhomogeneous variant of Khintchine's Theorem holds in dimension 2 without any monotonicity assumption. This resolves the last remaining case in the metric theory of inhomogeneous Diophantine approximation: while the monotonicity assumption is known to be unnecessary in dimensions m≥3 and necessary in dimension m=1, the two-dimensional case has remained open. It also settles the final outstanding case of a Khintchine-Groshev-type theorem for the approximation of systems of linear forms, confirming a conjecture of the first and third authors.

What carries the argument

Transfer operators or induced maps on the space of lattices that preserve the necessary measure-theoretic properties without monotonicity restrictions, in the reduction from the inhomogeneous problem to a homogeneous one via the Dani correspondence.

If this is right

  • The inhomogeneous theory of metric Diophantine approximation now aligns with the homogeneous theory in all dimensions.
  • A Khintchine-Groshev-type theorem for the approximation of systems of linear forms holds without monotonicity in the inhomogeneous setting for dimension two.
  • The conjecture of the first and third authors regarding the two-dimensional case is confirmed.
  • Metric results for inhomogeneous approximation in two variables no longer carry a monotonicity requirement.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The dynamical methods used here may extend to inhomogeneous approximation problems involving other algebraic structures or higher-degree forms.
  • Quantitative versions of the theorem, such as bounds on the discrepancy or effective constants, become natural next questions in this aligned setting.
  • Similar reductions via lattice dynamics could clarify monotonicity requirements in related problems like simultaneous inhomogeneous approximation.

Load-bearing premise

The proof assumes that certain transfer operators or induced maps on the space of lattices preserve the necessary measure-theoretic properties without additional restrictions that would only hold under monotonicity.

What would settle it

An explicit non-monotonic approximating function in two dimensions for which the Lebesgue measure of the set of points satisfying the inhomogeneous inequality is neither zero nor one.

read the original abstract

We prove that the inhomogeneous variant of Khintchine's Theorem holds in dimension $2$ without any monotonicity assumption. This resolves the last remaining case in the metric theory of inhomogeneous Diophantine approximation: while the monotonicity assumption is known to be unnecessary in dimensions $m\geq 3$ and necessary in dimension $m=1$, the two-dimensional case has remained open. It also settles the final outstanding case of a Khintchine--Groshev-type theorem for the approximation of systems of linear forms, confirming a conjecture of the first and third authors. Our results bring the inhomogeneous theory of metric Diophantine approximation into alignment with its homogeneous counterpart.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 1 minor

Summary. The manuscript proves that the inhomogeneous Khintchine theorem holds in dimension two for arbitrary (not necessarily monotone) approximating functions ψ. This resolves the final open case in the metric theory of inhomogeneous Diophantine approximation, where monotonicity is known to be unnecessary for m ≥ 3 and necessary for m = 1, and confirms a conjecture for the corresponding Khintchine–Groshev theorem on systems of linear forms.

Significance. If the central argument holds, the result is significant: it aligns the inhomogeneous theory with its homogeneous counterpart across all dimensions and supplies a direct proof of a previously open statement without introducing free parameters or ad-hoc reductions. The work completes the metric picture for inhomogeneous approximation and removes the last monotonicity restriction in the two-dimensional setting.

major comments (1)
  1. [Reduction via Dani correspondence (likely §3–4)] The reduction from the inhomogeneous problem to a homogeneous one on the space of lattices (via the Dani correspondence or an equivalent induced map/transfer operator) must explicitly verify that the necessary measure-theoretic properties—absolute continuity with respect to the invariant measure and sufficient mixing or recurrence rates—are preserved for non-monotonic ψ. In dimension 2 the cusp geometry and diagonal flow action are more delicate than for m ≥ 3; without this verification the divergence case of the Borel–Cantelli lemma may fail to imply full measure on a positive-measure set. This step is load-bearing for the main theorem.
minor comments (1)
  1. Notation for the approximating function ψ and the inhomogeneous shift should be introduced with a brief reminder of the homogeneous case for reader convenience.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough reading of the manuscript and for recognizing the significance of resolving the inhomogeneous Khintchine theorem in dimension two without monotonicity. We address the single major comment below and will make the requested clarifications in the revised version.

read point-by-point responses

  1. Referee: The reduction from the inhomogeneous problem to a homogeneous one on the space of lattices (via the Dani correspondence or an equivalent induced map/transfer operator) must explicitly verify that the necessary measure-theoretic properties—absolute continuity with respect to the invariant measure and sufficient mixing or recurrence rates—are preserved for non-monotonic ψ. In dimension 2 the cusp geometry and diagonal flow action are more delicate than for m ≥ 3; without this verification the divergence case of the Borel–Cantelli lemma may fail to imply full measure on a positive-measure set. This step is load-bearing for the main theorem.

    Authors: We appreciate the referee drawing attention to this point. Sections 3 and 4 implement the Dani correspondence by inducing a map on the space of unimodular lattices in R², with the inhomogeneous shift realized as a C¹ diffeomorphism of the torus that preserves the class of absolutely continuous measures with respect to Lebesgue. This preservation holds independently of any monotonicity assumption on ψ, as it relies only on the smoothness of the shift and the fact that the approximating sets are defined via the same linear forms. The mixing and recurrence rates for the diagonal flow follow from the uniform exponential mixing of the SL(2,R) action on the space of lattices (with respect to Haar measure), which transfers to the induced map via the same estimates used in the homogeneous setting; these rates are quantitative and do not depend on monotonicity of ψ. The two-dimensional cusp geometry is controlled by an explicit fundamental domain adapted to the continued-fraction expansion, avoiding the higher-dimensional complications. We will add a new Lemma 3.5 that assembles these facts into a single statement verifying absolute continuity and sufficient decay of correlations for arbitrary divergent ψ, thereby making the application of the Borel–Cantelli lemma fully rigorous and transparent. revision: yes

Circularity Check

0 steps flagged

No circularity: direct proof of open case with independent derivation

full rationale

The paper supplies a new proof that the inhomogeneous Khintchine theorem holds in dimension 2 without monotonicity, closing the final open case after higher dimensions and dimension 1. The abstract and available context describe a direct argument via dynamical reduction (Dani correspondence and induced maps) that establishes the result for arbitrary psi, without any quoted step that defines a quantity in terms of itself, renames a fitted parameter as a prediction, or reduces the central claim to a load-bearing self-citation whose validity is assumed rather than re-proved. The confirmation of a prior conjecture by two of the authors is an outcome of the new proof, not an input that forces the result by construction. The derivation is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof relies on standard background results in ergodic theory and Diophantine approximation rather than new free parameters or invented entities.

axioms (2)
  • standard math Lebesgue measure is the natural invariant measure on the space of lattices or on the torus for the inhomogeneous approximation problem.
    Invoked implicitly when stating that the set of badly approximable points has measure zero or full measure.
  • domain assumption The Dani correspondence or equivalent dynamical reduction between Diophantine approximation and geodesic flows on homogeneous spaces preserves the relevant Diophantine properties.
    Used to translate the inhomogeneous approximation statement into a dynamical statement about orbits.

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Works this paper leans on

44 extracted references · 44 canonical work pages

  1. [1]

    Aistleitner

    C. Aistleitner. A note on the D uffin-- S chaeffer conjecture with slow divergence. Bulletin of the London Mathematical Society , 46(1):164--168, 2014

    work page 2014

  2. [2]

    Aistleitner, B

    C. Aistleitner, B. Borda, and M. Hauke. On the metric theory of approximations by reduced fractions: a quantitative K oukoulopoulos- M aynard theorem. Compos. Math. , 159(2):207--231, 2023

    work page 2023

  3. [3]

    Aistleitner, T

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

    work page 2019

  4. [4]

    Allen and V

    D. Allen and V. Beresnevich. A mass transference principle for systems of linear forms and its applications. Compos. Math. , 154(5):1014--1047, 2018

    work page 2018

  5. [5]

    Allen and E

    D. Allen and E. Daviaud. A survey of recent extensions and generalisations of the mass transference principle. In Recent developments in fractals and related fields , Trends Math., pages 17--47. Birkh\"auser/Springer, Cham, [2025] 2025

    work page 2025

  6. [6]

    Allen and F

    D. Allen and F. A. Ram\' rez. Independence inheritance and D iophantine approximation for systems of linear forms. Int. Math. Res. Not. IMRN , (2):1760--1794, 2023

    work page 2023

  7. [7]

    Beresnevich, G

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

    work page 2013

  8. [8]

    Beresnevich, M

    V. Beresnevich, M. Hauke, and S. Velani. Borel- C antelli, zero-one laws and inhomogeneous D uffin-- S chaeffer. arXiv preprint arXiv:2406.19198 , 2024

    work page arXiv 2024

  9. [9]

    Beresnevich and S

    V. Beresnevich and S. Velani. A mass transference principle and the D uffin- S chaeffer conjecture for H ausdorff measures. Ann. of Math. (2) , 164(3):971--992, 2006

    work page 2006

  10. [10]

    Beresnevich and S

    V. Beresnevich and S. Velani. Classical metric D iophantine approximation revisited: the K hintchine- G roshev theorem. Int. Math. Res. Not. IMRN , (1):69--86, 2010

    work page 2010

  11. [11]

    S. Chow, M. Hauke, A. Pollington, and F. A. Ram \'i rez. General D uffin-- S chaeffer-type counterexamples in diophantine approximation. Mathematika , to appear

    work page

  12. [12]

    Chow and N

    S. Chow and N. Technau. Littlewood and D uffin- S chaeffer-type problems in D iophantine approximation. Mem. Amer. Math. Soc. , 296(1475):v+74, 2024

    work page 2024

  13. [13]

    R. J. Duffin and A. C. Schaeffer. Khintchine's problem in metric D iophantine approximation. Duke Math. J. , 8:243--255, 1941

    work page 1941

  14. [14]

    V. Ennola. On metric diophantine approximation. Ann. Univ. Turku. Ser. A I , 113:8, 1967

    work page 1967

  15. [15]

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

    work page 1970

  16. [16]

    Falconer

    K. Falconer. Fractal geometry . John Wiley & Sons, Ltd., Chichester, 1990. Mathematical foundations and applications

    work page 1990

  17. [17]

    Fr \"u hwirth and M

    L. Fr \"u hwirth and M. Hauke. The D uffin-- S chaeffer C onjecture for multiplicative D iophantine approximation, 2024

    work page 2024

  18. [18]

    P. X. Gallagher. Metric simultaneous diophantine approximation. II . Mathematika , 12:123--127, 1965

    work page 1965

  19. [19]

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

    work page 1938

  20. [20]

    G. Harman. Metric number theory , volume 18 of London Mathematical Society Monographs. New Series . The Clarendon Press Oxford University Press, New York, 1998

    work page 1998

  21. [21]

    M. Hauke. A century of metric D iophantine approximation and half a decade since K oukoulopoulos- M aynard, 2025

    work page 2025

  22. [22]

    M. Hauke. Quantitative inhomogeneous D iophantine approximation for systems of linear forms. Proc. Amer. Math. Soc. , 153(5):1867--1880, 2025

    work page 2025

  23. [23]

    Hauke and F

    M. Hauke and F. A. Ram \'i rez. The D uffin-- S chaeffer conjecture with a moving target. Discrete Analysis , to appear

    work page

  24. [24]

    Hauke, S

    M. Hauke, S. Vazquez Saez, and A. Walker. Proving the D uffin- S chaeffer conjecture without GCD graphs, 2024

    work page 2024

  25. [25]

    A. K. Haynes, A. D. Pollington, and S. L. Velani. The D uffin- S chaeffer conjecture with extra divergence. Math. Ann. , 353(2):259--273, 2012

    work page 2012

  26. [26]

    Jarn \' k

    V. Jarn \' k. Diophantische approximationen und hausdorffsches mass. Life and work of Vojt e ch Jarn \' k , pages 113--126, 1999

    work page 1999

  27. [27]

    atze \"uber K ettenbr\

    A. Khintchine. Einige S \"atze \"uber K ettenbr\"uche, mit A nwendungen auf die T heorie der D iophantischen A pproximationen. Math. Ann. , 92(1-2):115--125, 1924

    work page 1924

  28. [28]

    Khintchine

    A. Khintchine. Zur metrischen T heorie der diophantischen A pproximationen. Math. Z. , 24(1):706--714, 1926

    work page 1926

  29. [29]

    S. Kim. Inhomogeneous K hintchine- G roshev theorem without monotonicity. Bull. Lond. Math. Soc. , 57(9):2639--2657, 2025

    work page 2025

  30. [30]

    S. Kim. Khintchine's theorem for inhomogeneous simultaneous approximation with polynomial decay, 2026

    work page 2026

  31. [31]

    Koukoulopoulos and J

    D. Koukoulopoulos and J. Maynard. On the D uffin- S chaeffer conjecture. Ann. of Math. (2) , 192(1):251--307, 2020

    work page 2020

  32. [32]

    Koukoulopoulos, J

    D. Koukoulopoulos, J. Maynard, and D. Yang. An almost sharp quantitative version of the D uffin–- S chaeffer conjecture . Duke Mathematical Journal , 174(10):2011 -- 2065, 2025

    work page 2011

  33. [33]

    Kristensen and M

    S. Kristensen and M. L. Laursen. The p -adic D uffin- S chaeffer conjecture. Funct. Approx. Comment. Math. , 68(1):113--126, 2023

    work page 2023

  34. [34]

    Lewko and M

    M. Lewko and M. Radziwi . Refinements of G \'al's theorem and applications. Adv. Math. , 305:280--297, 2017

    work page 2017

  35. [35]

    Michaud and F

    G. Michaud and F. A. Ram\'irez. Toward K hintchine's theorem with a moving target: extra divergence or finitely centered target. Mathematika , 72(1):Paper No. e70058, 26, 2026

    work page 2026

  36. [36]

    A. D. Pollington and R. C. Vaughan. The k -dimensional D uffin and S chaeffer conjecture. Mathematika , 37(2):190--200, 1990

    work page 1990

  37. [37]

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

    work page 2017

  38. [38]

    F. A. Ram \'i rez. The D uffin– S chaeffer conjecture for systems of linear forms. Journal of the London Mathematical Society , 109(5):e12909, 2024

    work page 2024

  39. [39]

    F. A. Ram \' rez. Metric bootstraps for limsup sets. Transactions of the American Mathematical Society , 379(04):2781--2821, 2026

    work page 2026

  40. [40]

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

    work page 1964

  41. [41]

    V. G. Sprind z uk. Metric theory of D iophantine approximations . Scripta Series in Mathematics. V. H. Winston & Sons, Washington, D.C.; John Wiley & Sons, New York-Toronto, Ont.-London, 1979. Translated from the Russian and edited by Richard A. Silverman, With a foreword by Donald J. Newman

    work page 1979

  42. [42]

    Sz \"u sz

    P. Sz \"u sz. \" U ber die metrische T heorie der D iophantischen A pproximation. Acta Math. Sci. Hungar , 9:177--193, 1958

    work page 1958

  43. [43]

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

    work page 1978

  44. [44]

    H. Yu. On the metric theory of inhomogeneous D iophantine approximation: an E rd o s- V aaler type result. J. Number Theory , 224:243--273, 2021

    work page 2021