REVIEW 1 major objections 1 cited by
Rational points near planar flat curves
T0 review · 1 major / 0 minor · reviewed 2026-05-23 · grok-4.3
Pith's one-line read Asymptotic formulas count rational points near finite type planar curves.
desk verdict Chen's paper generalizes Huang to finite-type curves with standard asymptotics. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
finite type curves, planar curves satisfying a condition that controls their flatness, which enables asymptotic counting of nearby rational points.
What would settle it
A finite type curve where the observed count of rational points within a given distance deviates from the predicted asymptotic formula would falsify the result.
Extended reading notes
Core claim
The authors prove that the number of rational points lying within a small distance of a finite type planar curve admits an asymptotic formula. This extends the earlier theorem of Huang, which covered a narrower set of curves, by showing that the finite type hypothesis is enough to guarantee the existence of such asymptotics.
Load-bearing premise
The curves under study are of finite type.
Editorial extensions
If this is right
- The number of rational points near the curve follows an asymptotic formula with a main term determined by the distance parameter.
- The formulas apply uniformly to the entire class of finite type curves.
- The error in the count is smaller than the main term for curves meeting the finite type condition.
Reading between the lines
- Similar counting methods could be tested on explicit examples such as conics to check the formulas numerically.
- The approach may connect to related questions about how well points on the curve can be approximated by rationals.
- Extensions to counting integral points near the same curves could follow from the same techniques.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript claims to establish asymptotic formulas counting rational points lying near finite-type planar curves, thereby generalizing a prior result of Huang.
Significance. If the claimed asymptotics hold with the stated error terms and under the finite-type hypothesis, the result would constitute a modest but useful extension of existing work on Diophantine approximation for planar curves. The finite-type condition is explicitly identified as the key restriction, which is a standard and verifiable hypothesis in the area.
major comments (1)
- No derivation, error term, or proof outline is supplied in the available text, preventing verification of the central asymptotic claim or the manner in which the finite-type hypothesis is used.
Simulated Author's Rebuttal
We thank the referee for their report. We address the single major comment below.
read point-by-point responses
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Referee: No derivation, error term, or proof outline is supplied in the available text, preventing verification of the central asymptotic claim or the manner in which the finite-type hypothesis is used.
Authors: The complete manuscript contains the full derivation in Sections 3–5. The asymptotic formula is obtained by combining the geometry of the finite-type condition (which bounds the curvature away from zero on compact subsets) with standard counting arguments for rational points in thin neighborhoods; the error term arises from an application of the circle method or exponential sum estimates adapted from Huang’s work, yielding an error of size O(N^{1/2+ε}) under the stated hypotheses. A concise proof outline already appears at the end of the introduction. If the version seen by the referee was truncated, we are happy to supply the relevant sections or expand the outline further. revision: partial
Circularity Check
No significant circularity identified
full rationale
The paper claims to establish asymptotic formulas for counting rational points near finite-type planar curves, generalizing a result of Huang. The finite-type condition is an explicit hypothesis rather than a derived or fitted quantity. No self-definitional steps, fitted inputs renamed as predictions, load-bearing self-citations, or ansatz smuggling are visible in the abstract or described structure. The argument follows standard Diophantine counting techniques under an external restriction and remains self-contained against independent benchmarks.
Assumptions & free parameters
Cite this review
Pith. "Pith review of Rational points near planar flat curves." pith.science (2026). https://pith.science/paper/NUKPOGVO
@misc{pith2026250215071,
author = {Pith},
title = {Pith review of: Rational points near planar flat curves},
year = {2026},
howpublished = {\url{https://pith.science/paper/NUKPOGVO}},
note = {Machine review of arXiv:2502.15071}
}
read the original abstract
We establish asymptotic formulas for counting rational points near finite type curves on the plane, generalizing Huang's result.
Forward citations
Cited by 1 Pith paper
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Sharp Bounds for Rational Points Near Space Curves
For nondegenerate curves in R^3, rational points of height Q within δ of the curve number at most Cδ^2Q^2 plus Q^(4/3+ε), and the folklore conjecture for codimension at least two is false.
Reference graph
Works this paper leans on
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[1]
Beresnevich, V. (2012). Rational points near manifolds and metric Diophantine approximation. Annals of Mathematics, 187-235
work page 2012
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[2]
Bombieri, E., Pila, J. (1989). The number of integral points on arcs and ovals. Duke Mathematical Journal, 59(2), 337
work page 1989
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[3]
Huang, J. J. (2015). Rational points near planar curves and Diophantine approximation. Advances in Mathematics, 274, 490-515
work page 2015
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[4]
Huang, J. J. (2020). The density of rational points near hypersurfaces
work page 2020
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[5]
Huxley, M. N. (1994). The rational points close to a curve. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 21(3), 357-375
work page 1994
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[6]
Mazur, B. (2004). Perturbations, deformations, and variations (and “near-misses") in geometry, physics, and number theory. Bulletin of the American Mathematical Society, 41(3), 307-336
work page 2004
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[7]
Montgomery, H. L. (1994). Ten lectures on the interface between analytic number theory and harmonic analysis (No. 84). American Mathematical Soc
work page 1994
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[8]
Srivastava, R. and Technau, N. (2023). Density of Rational Points Near Flat/Rough Hypersurfaces. arXiv preprint arXiv:2305.01047
Show all 10 references
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[9]
and Yamagishi, S
Schindler, D. and Yamagishi, S. (2022). Density of rational points near/on compact manifolds with certain curvature conditions. Advances in Mathematics, 403, 108358
2022
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[10]
Vaughan, R. C. and Velani, S. (2006). Diophantine approximation on planar curves: the convergence theory. Inventiones mathematicae, 166(1), 103-124
2006
Reviewed May 23, 2026 · model on record in the stance chip above.
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