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arxiv: 2502.15071 · v2 · pith:NUKPOGVOnew · submitted 2025-02-20 · 🧮 math.NT · math.CA

Rational points near planar flat curves

Mingfeng Chen This is my paper

Pith reviewed 2026-05-23 02:05 UTC · model grok-4.3

classification 🧮 math.NT math.CA
keywords rational pointsfinite type curvesplanar curvesasymptotic formulasDiophantine approximationnumber theory
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The pith

Asymptotic formulas count rational points near finite type planar curves.

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

The paper establishes asymptotic formulas for counting rational points near finite type curves on the plane. This generalizes Huang's result to a larger class of curves. A reader would care because such formulas give precise estimates on how densely rational points can approximate points on the curve. The work shows that the finite type condition suffices for the asymptotic to hold with a controlled error term.

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.

What carries the argument

finite type curves, planar curves satisfying a condition that controls their flatness, which enables asymptotic counting of nearby rational points.

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.

Where Pith is reading between the lines

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

  • 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.

Load-bearing premise

The curves under study are of finite type.

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.

read the original abstract

We establish asymptotic formulas for counting rational points near finite type curves on the plane, generalizing Huang's result.

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 / 0 minor

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)
  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

1 responses · 0 unresolved

We thank the referee for their report. We address the single major comment below.

read point-by-point responses

  1. 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

0 steps flagged

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.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract alone.

pith-pipeline@v0.9.0 · 5512 in / 809 out tokens · 36874 ms · 2026-05-23T02:05:17.028785+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

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    work page 2015

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    Huang, J. J. (2020). The density of rational points near hypersurfaces

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    near-misses

    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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    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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    and Technau, N

    Srivastava, R. and Technau, N. (2023). Density of Rational Points Near Flat/Rough Hypersurfaces. arXiv preprint arXiv:2305.01047

    work page arXiv 2023

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    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

    work page 2022

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    Vaughan, R. C. and Velani, S. (2006). Diophantine approximation on planar curves: the convergence theory. Inventiones mathematicae, 166(1), 103-124

    work page 2006