Sharp Bounds for Rational Points Near Space Curves

Andreas Seeger; Mingfeng Chen; Niclas Technau; Rajula Srivastava

REVIEW 1 major objections 4 minor 33 references

Sharp Bounds for Rational Points Near Space Curves

T0 review · 1 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proves essentially sharp upper bounds for rational points near non-degenerate space curves in R^3 and shows that a folklore counting conjecture fails for codimension at least two.

desk verdict Important, likely correct result with a real but probably fixable gap in Lemma 8.3; worth refereeing carefully. read the letter →

arxiv 2608.09009 v1 pith:HW7GGS2T submitted 2026-08-10 math.NT math.CA

Mingfeng Chen , Andreas Seeger , Rajula Srivastava , Niclas Technau This is my paper · ORCID

classification math.NTmath.CA MSC 11J8311J1311J2542B20

keywords rationalpointsnearcurvesDiophantineapproximationspacenonvanishingcurvatureandtorsionstationaryphaseslowdecayconetangentdevelopablesurfacemajorarcobstruction

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

This paper counts rational points a/q of height q≤Q lying within distance δ/q of a smooth space curve in $R^{3}$ with nonvanishing curvature and torsion. It proves the essentially optimal upper bound N_C(δ,Q) ≤ C δ²Q² + O_ε($Q^{{4/3+ε}}$), together with an asymptotic for a smoothed count whose leading term is c δ²Q². It also exhibits a major-arc obstruction: for cylinders over the moment curve in codimension at least two, the number of near rational points is at least $δ^{{1/2}}$$Q^{{n−m+1/2}}$ in a range where previous conjectures predicted a smaller number. If correct, the results settle the sharp order of rational points near 3-nondegenerate space curves and disprove the folklore conjecture except in the hypersurface case.

What carries the argument

The central object is the slow-decay (binormal) cone Γ = {θ : γ′(t)·θ = γ″(t)·θ = 0 for some t}, together with the function u(ξ) = φ′_ξ(θ_2(ξ)), where θ_2 locates the minimum of the phase derivative; u(ξ) measures signed distance to Γ. The argument decomposes the nonzero frequencies dyadically by |u(ξ)| ≍ $2^{{−2ℓ}}$; near Γ, the phase has two critical points separated by ≍ $2^{{−ℓ}}$, and Theorem 4.12 supplies a stationary-phase expansion with uniformly controlled remainders at every dyadic scale. This reduces each dyadic piece of the original count to a dual count of integer frequencies near the tangent developable S*, which is then bounded using a planar rational-point theorem and an involutive Legendre-transform reduction back to the original curve.

What would settle it

Verify Lemma 8.3’s decomposition for the normalized moment curve γ(t)=(t,t²/2,t³/6) with λ=$2^{{3ℓ}}$ and u(ξ)≍$2^{{−2ℓ}}$; if the stated equality I(λ,ξ)=I^+(λ,ξ,ℓ)+I^−(λ,ξ,ℓ)+O($λ^{{−N}}$) is not reproduced, or the claimed separation of the two critical points from the bump width fails, then Theorem 4.12—and hence the proof of the upper bound—is not currently supported. A separate computational test would be to count rational points for a specific nondegenerate curve at δ≈$Q^{{−1/3}}$ and check whether the count follows δ²Q² plus an error of size $Q^{{4/3+ε}}$.

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Extended reading notes

Core claim

On the paper's own terms: the number of rational points of height ≤Q in a δ/q-neighborhood of a compact C∞ curve in $R^{3}$ with nonvanishing curvature and torsion is O_ε(δ²Q² + $Q^{{4/3+ε}}$), and for δ ≥ $Q^{{-1/3+ε}}$ the count is comparable to δ²Q². The proof passes from the original counting problem to oscillatory sums, then to a dual counting problem near a tangent developable surface S* in frequency space. The key mechanism is a dyadic stationary-phase expansion near the slow-decay cone Γ, where the phase has two nearby critical points and the Fourier transform of the curve decays slowest. The same mechanism reveals a previously hidden major-arc obstruction: cylinders over moment curves have far more near rational points than the conjectured δ^m $Q^{{n−m+1}}$, so the codimension-m≥2 form of the conjecture fails.

Load-bearing premise

The load-bearing premise is that Theorem 4.12’s stationary-phase expansion, with its O($λ^{{−N}}$) remainder uniform in the dyadic scale ℓ, is a valid theorem; because the displayed identity in Lemma 8.3 used in its proof is false as written, the upper bound stands only if that lemma’s intended separation argument repairs the step.

Editorial extensions

If this is right

For every compact C∞ curve in R^3 with nonvanishing curvature and torsion, N_C(δ,Q) ≤ C δ²Q² + O_ε(Q^{4/3+ε}); in the range δ ≥ Q^{-1/3+ε}, the two-sided bound c δ²Q² ≤ N_C(δ,Q) ≤ C δ²Q² holds.
The folklore conjecture (Conjecture 3.1) is false for codimension m≥2: for cylinders over the moment curve, N_{M_{m,n}}(δ,Q) ≳ δ^{1/2}Q^{n−m+1/2} whenever A Q^{ε−1} ≤ δ ≤ 1/2.
Because the conjectured uniform bound fails in this range, sharp counting near a special manifold cannot by itself imply the convergence case of Khintchine's theorem; generic/special decompositions are necessary rather than merely convenient.
The earlier estimate N_C(δ,Q) ≲ δ²Q² + Q^{8/5}(log Q)^{4/5} for such curves is improved to an error term of size Q^{4/3+ε}.
The smooth asymptotic N_{Ω,C}(δ,Q) = c_Ω δ²Q² + O_ε(δ^{4/3−ε}Q^{5/3+3ε} + δ^{1/2−ε}Q^{3/2+ε}) holds for δ in (Q^{ε−1/2}, 1/2), giving an explicit leading constant.

Reading between the lines

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

A natural next step is to extend the dyadic slow-decay-cone analysis to nondegenerate curves in R^n; the structure of the proof suggests dimension-dependent exponents and analogous major-arc obstructions, although the paper proves only the case n=3.
The failure of the codimension≥2 conjecture suggests that the sharp exponent for rational points near a manifold is controlled by the slowest Fourier-decay directions of its surface measure rather than by codimension alone; this reading goes beyond what the paper states explicitly.
A testable outgrowth would be to count rational points near the tangent developable S* directly for a cubic model curve and compare the resulting exponent with the dual-count estimate (5.8); agreement would confirm that the Legendre-involution step is lossless.

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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

This is a strong paper. It settles the order of magnitude for rational points near nondegenerate space curves in R^3: the upper bound in Theorem 1.2 improves Huang's Q^{8/5} (up to logs) to Q^{4/3+epsilon}, and the lower bound in Theorem 1.1 for the moment curve cylinders disproves Huang's conjecture for codimension at least two. The lower bound is elementary and sound, and the overall architecture of the upper-bound proof is coherent: reduce to a fixed-scale perturbation of the moment curve, decompose frequencies by distance to the binormal cone, pass to a Legendre dual counting problem, and invoke the Vaughan–Velani planar theorem for the final count. The paper is honest about what is new and credits the relevant literature, including Huang, Beresnevich–Yang, Hickman–Srivastava, and Schindler–Yamagishi.

The soft spot is real. Lemma 8.3, which feeds directly into Theorem 4.12, the stationary-phase expansion used to pass to the dual counting problem, claims the identity (8.5):

1 = eta((t-theta^-)/alpha) + (1-eta((t-theta^+)/alpha))(1-eta((t-theta^-)/alpha)) + eta((t-theta^+)/alpha).

Expanding the middle term gives an extra eta((t-theta^-)/alpha)eta((t-theta^+)/alpha), so the identity is false unless the two bumps have disjoint supports. The missing overlap term is not included in the remainder r_l and is not estimated. Since Lemma 8.3 is used in Lemma 8.4 and hence in Theorem 4.12, which is invoked in the proof of Proposition 5.3, the main upper bound currently rests on an unproved estimate. The gap is localized and plausibly fixable: Lemma 4.10 gives |theta^+ - theta^-| ~ 2^{-ell}, and under the scale condition (4.54) one has alpha = lambda_ell^{-1/2+epsilon} = o(2^{-ell}), so the supports are in fact disjoint; the separation argument is just not written. Still, as the paper stands, Theorem 4.12 lacks a complete proof.

This is a conditional accept situation, not a reject. The surrounding argument gives strong evidence that the intended approach works, and the flaw is confined to one step. A few smaller presentation issues (several omitted 'straightforward' modifications, a slightly overbroad abstract claim about folklore conjectures being incorrect for codimension at least two when the counterexample is exhibited for specific cylinders) do not change the picture. The paper deserves a serious referee, and I would send it out with the expectation that Section 8 needs fixing. The reader's verdict of CONDITIONAL with moderate confidence matches my own reading.

Referee Report

1 major / 4 minor

Summary. The paper studies the counting function N_C(δ,Q) for rational points of height at most Q lying within δ/q of a compact C∞ curve C⊂R^3 with nonvanishing curvature and torsion. Its main upper bound, Theorem 1.2, asserts N_C(δ,Q) ≤ Cδ^2Q^2 + O_ε(Q^{4/3+ε}), which improves Huang's earlier Q^{8/5} bound and matches the lower bound up to endpoints in the range δ > Q^{-1/3+ε}. The paper also proves Theorem 1.1, an elementary lower bound N_{M_{m,n}}(δ,Q) ≳ δ^{1/2} Q^{n-m+1/2} for cylinders over the moment curve, which contradicts Huang's Conjecture 3.1 for codimension m≥2. The proof strategy combines a truncated Fourier expansion, a decomposition of frequencies according to distance to the binormal cone Γ, a quantitative stationary phase expansion (Theorem 4.12), and a reduction to a dual counting problem for the tangent developable surface S*; the dual count is ultimately bounded using the external Vaughan–Velani planar theorem.

Significance. If the proof is completed, the results are significant. Theorem 1.2 gives the essentially sharp exponent for rational points near nondegenerate space curves in R^3, and Theorem 1.1 provides a clean, elementary counterexample to a plausible general conjecture, with the 'major arc' obstruction clearly identified. The paper also introduces a reusable structural framework: dyadic decomposition near the slow-decay cone, a quantitative stationary phase expansion with uniform remainders, and a Legendre-dual counting argument adapted to a nonsmooth dual variety. The lower-bound argument in Section 2 is self-contained and checkable. The upper-bound argument is not circular: the dual counting problem is reduced back to the original curve count, but that count is then bounded using the external Vaughan–Velani planar theorem rather than the paper's own results. These strengths make the paper worth pursuing; however, one load-bearing proof step in the stationary phase section is currently incorrect as written.

major comments (1)

[Section 8, Lemma 8.3, Eq. (8.5)] The displayed partition identity is algebraically false. Expanding (1−η((t−θ+1)/α))(1−η((t−θ−1)/α)) gives 1 − η+ − η− + η+η−, so the right-hand side of (8.5) equals 1 + η((t−θ−1)/α)η((t−θ+1)/α), not 1. The term rℓ(t,ξ) defined immediately afterward omits this cross term, and no estimate for it is supplied. This is load-bearing because Lemma 8.3 is used in the proof of Theorem 4.12, and Theorem 4.12 is invoked in Lemma 6.2 to pass to the dual counting problem in Proposition 5.3; thus the main upper bound currently rests on an unproved overlap estimate. The gap appears fixable: Lemma 4.10 gives |θ+1−θ−1| ≍ 2^{−ℓ}, and condition (4.54) implies α = λ_ℓ^{−1/2+ε} = o(2^{−ℓ}) for large λ, so the supports of the two η factors are disjoint and the cross term vanishes. However, this separation argument is not written, and the displayed algebra as stated is wrong.

minor comments (4)

[Theorem 4.12 and Lemma 8.4] Theorem 4.12 states the remainder amplitude as W^σ_ℓ((λ2^{−ℓ})^ε, ξ), while Lemma 8.4 concludes with A^σ_ℓ(λ_ℓ^{2ε}, ξ). These two normalizations should be reconciled; since ε is arbitrary, the discrepancy is harmless, but as written the notation is inconsistent.
[Section 2, Lemmas 2.1 and 2.2] The notation N_{C_m} in Lemma 2.1 should be N̄_{C_m}, the pair count, because the proof counts pairs (a,q) before the Möbius-inversion step in Lemma 2.2; as written, the statement and proof use different counting functions.
[Section 2, display (2.1)] Expressions such as √δQ are typeset ambiguously; $\sqrt{\delta Q}$ should be used throughout so that it is not read as $(\sqrt{\delta})Q$.
[Section 8, proof of Lemma 8.2] The sentence 'the implicit constant depends on φ, a and N' should specify whether the estimate is uniform in ξ and ℓ; the argument appears to give uniformity, but the statement should say so explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the upper bound is closed through the external Vaughan–Velani theorem, and the stationary-phase ingredient is proved in Section 8.

full rationale

The claimed derivation is not circular. The main upper bound (Theorem 1.2) is deduced from the smoothed asymptotic Theorem 4.2, whose proof (Section 5) decomposes the counting function by Fourier expansion (equations (4.13)–(4.20)) into a main term and oscillatory remainders, controlled by Propositions 5.1–5.4. The key technical ingredient, Theorem 4.12, is proved self-contained in Section 8 via Lemmas 8.1–8.4; although Lemmas 4.5, 4.7 and 4.10 cite [4], the paper reproduces their proofs in the text, so those citations are not load-bearing. The dual counting estimate Proposition 5.4 is the only point where the argument returns to the original curve count: Lemma 7.6 bounds the exponential sums E by N_{\Omega_\ell^*,C}(Q_*^{-1},2^s), but the proof then bounds this quantity by dropping the third-coordinate factor and invoking the external Vaughan–Velani planar theorem (Theorem 3.7), not the paper's own Theorem 4.2. Thus the return to the curve is an application of an independent external benchmark. The lower bound Theorem 1.1 uses only elementary Möbius inversion and is independent of the upper-bound machinery. No parameter is fitted to the target count, and no claimed prediction coincides with an input by construction. The only overlap-author citation supplying a result is [26] for the lower bound in Corollary 1.3, which is not needed for Theorem 1.2. The algebraic issue in Lemma 8.3 (identity (8.5)) is a correctness gap in the stationary-phase proof, not a circularity.

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

The paper introduces no empirically fitted parameters and no new postulated physical entities. The binormal cone Γ, the tangent developable dual surface, and the dual counting functions N* are technical constructs defined explicitly from the given curve. The auxiliary constants ε0,...,ε3,k0,D0,A are proof variables chosen through inequalities, not tuned to data.

assumptions (5)

standard math Poisson summation formula
Used in Lemmas 3.1 and 3.2 to expand the smoothed counting function and evaluate geometric sums in the denominator variable.
standard math Stationary phase with parameters (Theorem 3.6)
Cited to [10,16,28] and used throughout Section 8 to obtain oscillatory asymptotics for the curve's Fourier transform.
standard math Vaughan–Velani planar counting theorem (Theorem 3.7)
External sharp bound for rational points near planar curves with nonvanishing curvature; used in Lemmas 4.14, 7.6 and Proposition 5.4.
standard math Möbius inversion and the identity sum_{d|n} μ(d) = 1_{n=1}
Used in Lemma 2.2 and Theorem 1.1 to pass from pairs (a,q) to coprime pairs; cited to Hardy and Wright [13].
standard math Van der Corput's lemma (Lemma 6.1)
Used to bound oscillatory integrals in Proposition 5.2; cited to Stein [31].

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Pith. "Pith review of Sharp Bounds for Rational Points Near Space Curves." pith.science (2026). https://pith.science/paper/HW7GGS2T

@misc{pith2026260809009,
  author       = {Pith},
  title        = {Pith review of: Sharp Bounds for Rational Points Near Space Curves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HW7GGS2T}},
  note         = {Machine review of arXiv:2608.09009}
}

abstract

Let $Q\geq 1$ be large, and $\delta \in(0,1)$ be small. Denote by $\mathcal C \subset \mathbb R^3$ a sufficiently smooth curve with non-vanishing curvature and torsion. How many rational points $\mathbf{a}/q$ of height $q\in[Q,2Q]$ are $\delta/q$-near $\mathcal C$? This manuscript provides an essentially optimal answer. We show that the folklore conjectures are incorrect for certain manifolds with codimension $\ge 2$, including the moment curve $(t,t^2,t^3)$. The reason is a hitherto hidden `major arc' type obstruction. We also establish matching upper bounds (up to endpoints). Our argument combines purely Fourier analytical techniques with the planar counting results by Vaughan--Velani.

Figures

Figures reproduced from arXiv: 2608.09009 by the authors.

**Figure 1.** From left to right: when ξ ∈ S1, then ϕξ has no critical points. If ξ ∈ S2, then there are either two well-separated or a single critical point. If ξ ∈ S3 (drawn in red), then ϕξ here two (possibly closely) clustering critical points which mimic the behavior of one critical point with order two. The following lemma describes how NC(δ, Q) can be expressed as an average over oscillatory integrals. Lemma 4.3. Recall ω1… view at source ↗

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Reference graph

Works this paper leans on

33 extracted references · 31 canonical work pages

[1]

J. F. Adams, Peter D. Lax, and Ralph S. Phillips,On matrices whose real linear combinations are non- singular, Proc. Amer. Math. Soc.16(1965), 318–322. Correction in Proc. Amer. Math. Soc. 17 (1966) 945–947

work page 1965
[2]

Theresa C. Anderson, Laura Cladek, Malabika Pramanik, and Andreas Seeger,Spherical means on the Heisenberg group: stability of a maximal function estimate, Journal d’Analyse Mathématique (2021), 1–28

work page 2021
[3]

G. I. Arkhipov, V. N. Chubarikov, and A. A. Karatsuba,Trigonometric sums in number theory and analysis, De Gruyter Expositions in Mathematics, vol. 39, Walter de Gruyter GmbH & Co. KG, Berlin,

work page
[4]

Math.393(2021), Paper No

David Beltran, Shaoming Guo, Jonathan Hickman, and Andreas Seeger,Sobolev improving for averages over curves inR4, Adv. Math.393(2021), Paper No. 108089, 85

work page 2021
[5]

Victor Beresnevich,Rational points near manifolds and metric Diophantine approximation, Annals of Mathematics (2012), 187–235. 46

work page 2012
[6]

Victor Beresnevich, Detta Dickinson, and Sanju Velani,Diophantine approximation on planar curves and the distribution of rational points, Ann. of Math. (2)166(2007), no. 2, 367–426. With an Appendix II by R. C. Vaughan

work page 2007
[7]

Victor Beresnevich and Lei Yang,Khintchine’s theorem and diophantine approximation on manifolds, Acta Mathematica231(2023), no. 1, 1–30

work page 2023
[8]

Rational points near planar flat curves

Mingfeng Chen,Rational points near planar flat curves. arXiv preprint arXiv:2502.15071

work page Pith review arXiv

Show all 33 references

[9]

forthcoming

Mingfeng Chen, Rajula Srivastava, and Niclas Technau,Rational points near cylindrical hypersurfaces. forthcoming
[10]

J. J. Duistermaat,Fourier integral operators, Modern Birkhäuser Classics, Birkhäuser/Springer, New York, 2011. Reprint of the 1996 edition [MR1362544], based on the original lecture notes published in 1973 [MR0451313]

2011
[11]

Arthur Erdelyi, W Magnus, F Oberhettinger, and FG Tricomi,Higher transcendental functions. Vol. II. Melbourne, FL, Robert E. Krieger Publishing Co. Inc., 1981

1981
[12]

Reine Angew

Allan Greenleaf and Andreas Seeger,Fourier integral operators with fold singularities, J. Reine Angew. Math.455(1994), 35–56

1994
[13]

G. H. Hardy and E. M. Wright,An introduction to the theory of numbers, Fifth, The Clarendon Press, Oxford University Press, New York, 1979

1979
[14]

14, rnaf200

Jonathan Hickman and Rajula Srivastava,Counting integral points near space curves: A Fourier analytic approach, International Mathematics Research Notices2025(2025), no. 14, rnaf200

2025
[15]

Jonathan Hickman, Rajula Srivastava, and James Wright,Counting rational points near manifolds: a refined estimate, a conjecture and a variant, arXiv preprint arXiv:2512.23204 (2025)

2025
[16]

I, Acta Math.127(1971), no

Lars Hörmander,Fourier integral operators. I, Acta Math.127(1971), no. 1-2, 79–183

1971
[17]

127, Birkhäuser Boston, Inc., Boston, MA, 1994

,Notions of convexity, Progress in Mathematics, vol. 127, Birkhäuser Boston, Inc., Boston, MA, 1994

1994
[18]

Ann.374(2019), no

Jing-Jing Huang,Integral points close to a space curve, Math. Ann.374(2019), no. 3-4, 1987–2003

2019
[19]

J.169(2020), no

,The density of rational points near hypersurfaces, Duke Math. J.169(2020), no. 11, 2045–2077

2020
[20]

1, 113–163

,Extremal affine subspaces and Khintchine-Jarník type theorems, Geometric and Functional Anal- ysis34(2024), no. 1, 113–163

2024
[21]

3, 357–375

Martin Neil Huxley,The rational points close to a curve, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze21(1994), no. 3, 357–375

1994
[22]

13, Clarendon Press, 1996

,Area, lattice points, and exponential sums, Vol. 13, Clarendon Press, 1996

1996
[23]

Smith,A Bessel function multiplier, Proc

Daniel Oberlin and Hart F. Smith,A Bessel function multiplier, Proc. Amer. Math. Soc.127(1999), no. 10, 2911–2915

1999
[24]

Phong and Elias M

Duong H. Phong and Elias M. Stein,Radon transforms and torsion, Internat. Math. Res. Notices4 (1991), 49–60

1991
[25]

Malabika Pramanik and Andreas Seeger,Lp regularity of averages over curves and bounds for associated maximal operators, Amer. J. Math.129(2007), no. 1, 61–103

2007
[26]

arXiv preprint arXiv:2310.03867

Damaris Schindler, Rajula Srivastava, and Niclas Technau,Rational points near manifolds, homogeneous dynamics, and oscillatory integrals. arXiv preprint arXiv:2310.03867

arXiv
[27]

Damaris Schindler and Shuntaro Yamagishi,Density of rational points near/on compact manifolds with certain curvature conditions, Advances in Mathematics403(2022), 108358

2022
[28]

210, Cambridge University Press, 2017

Christopher D Sogge,Fourier integrals in classical analysis, Vol. 210, Cambridge University Press, 2017

2017
[29]

Rajula Srivastava,Counting rational points in non-isotropic neighborhoods of manifolds, Advances in Mathematics478(2025), 110394

2025
[30]

arXiv preprint arXiv:2305.01047, to appear in Mem

Rajula Srivastava and Niclas Technau,Density of rational points near flat/rough hypersurfaces. arXiv preprint arXiv:2305.01047, to appear in Mem. Amer. Math. Soc

arXiv
[31]

Elias M Stein and Timothy S Murphy,Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton University Press, 1993

1993
[32]

1, 103–124

Robert C Vaughan and Sanju Velani,Diophantine approximation on planar curves: the convergence theory, Inventiones mathematicae166(2006), no. 1, 103–124. Glossary of notation General conventions. 47 Q, δ, εRespectively, the denominator/height scale, the thickness parame- ter, a...

2006
[2004]

Translated from the 1987 Russian original

1987

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

J. F. Adams, Peter D. Lax, and Ralph S. Phillips,On matrices whose real linear combinations are non- singular, Proc. Amer. Math. Soc.16(1965), 318–322. Correction in Proc. Amer. Math. Soc. 17 (1966) 945–947

work page 1965

[2] [2]

Theresa C. Anderson, Laura Cladek, Malabika Pramanik, and Andreas Seeger,Spherical means on the Heisenberg group: stability of a maximal function estimate, Journal d’Analyse Mathématique (2021), 1–28

work page 2021

[3] [3]

G. I. Arkhipov, V. N. Chubarikov, and A. A. Karatsuba,Trigonometric sums in number theory and analysis, De Gruyter Expositions in Mathematics, vol. 39, Walter de Gruyter GmbH & Co. KG, Berlin,

work page

[4] [4]

Math.393(2021), Paper No

David Beltran, Shaoming Guo, Jonathan Hickman, and Andreas Seeger,Sobolev improving for averages over curves inR4, Adv. Math.393(2021), Paper No. 108089, 85

work page 2021

[5] [5]

Victor Beresnevich,Rational points near manifolds and metric Diophantine approximation, Annals of Mathematics (2012), 187–235. 46

work page 2012

[6] [6]

Victor Beresnevich, Detta Dickinson, and Sanju Velani,Diophantine approximation on planar curves and the distribution of rational points, Ann. of Math. (2)166(2007), no. 2, 367–426. With an Appendix II by R. C. Vaughan

work page 2007

[7] [7]

Victor Beresnevich and Lei Yang,Khintchine’s theorem and diophantine approximation on manifolds, Acta Mathematica231(2023), no. 1, 1–30

work page 2023

[8] [8]

Rational points near planar flat curves

Mingfeng Chen,Rational points near planar flat curves. arXiv preprint arXiv:2502.15071

work page Pith review arXiv