Sharp Bounds for Rational Points Near Space Curves

Abstract

Let Q≥ 1 be large, and δ∈(0,1) be small. Denote by 𝒞 ⊂ ℝ³ a sufficiently smooth curve with non-vanishing curvature and torsion. How many rational points a/q of height q∈[1, Q] are δ/q-near 𝒞? This manuscript provides an essentially optimal answer, thereby addressing a problem stated by Beresnevich and Kleinbock, for space curves. We show that the folklore conjectures are incorrect for certain manifolds with codimension ≥ 2, including the moment curve (t,t²,t³). 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 and Velani.

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BibTeX RIS

BibTeX

@misc{chen2026sharp,
  title = {{Sharp Bounds for Rational Points Near Space Curves}},
  author = {Mingfeng Chen and Andreas Seeger and Rajula Srivastava and Niclas Technau},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.09009},
  eprint = {2608.09009},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.09009},
}

RIS

TY  - UNPB
TI  - Sharp Bounds for Rational Points Near Space Curves
AU  - Mingfeng Chen
AU  - Andreas Seeger
AU  - Rajula Srivastava
AU  - Niclas Technau
PY  - 2026
DA  - 2026/09/05
PB  - arXiv
DO  - 10.48550/arxiv.2608.09009
UR  - https://arxiv.org/abs/2608.09009
AB  - Let Q≥ 1 be large, and δ∈(0,1) be small. Denote by 𝒞 ⊂ ℝ^3 a sufficiently smooth curve with non-vanishing curvature and torsion. How many rational points a/q of height q∈[1, Q] are δ/q-near 𝒞? This manuscript provides an essentially optimal answer, thereby addressing a problem stated by Beresnevich and Kleinbock, for space curves. We show that the folklore conjectures are incorrect for certain manifolds with codimension ≥ 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 and Velani.
ER  -

Abstract

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