Counterexamples for Rational Points Near Curves

Abstract

Let δ∈ [0,1/2] and Q≥ 1. Given a compact C^∞-curve 𝒞 in ℝⁿ, denote by N_𝒞(δ, Q) the number of pairs (a, q)∈ ℤⁿ× [Q/2,Q] such that the rational point a/q is δ/q-close to 𝒞. We investigate the range of δ in terms of Q which is necessary for the heuristic N_𝒞(δ, Q)asympδⁿ⁻¹Q² to be correct for the moment curve. For a nondegenerate curve 𝒞⊂ ℝⁿ and for sufficiently large Q, Hickman and the second author previously established that N_𝒞(δ, Q)≲_ν δⁿ⁻¹Q² for all δ∈[Q^α(n+1)+ν,1/2) and any ν>0. Here α(n)=−4n⁻²+O(n⁻³) was explicitly computed. We show that this range is surprisingly close to being sharp in the asymptotic sense as n→∞, and can at most be extended to δ∈[Q^α(n),1/2). In particular, the sharp exponent can only be about O(n⁻³) better, even though it was widely believed before that an improvement of O(n⁻¹) should be possible. The complementary upper bound in this remaining range has been recently established by Gan--Guo--Oh.

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BibTeX RIS

BibTeX

@misc{chen2026counterexamples,
  title = {{Counterexamples for Rational Points Near Curves}},
  author = {Mingfeng Chen and Rajula Srivastava and Niclas Technau},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.02443},
  eprint = {2610.02443},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.02443},
}

RIS

TY  - UNPB
TI  - Counterexamples for Rational Points Near Curves
AU  - Mingfeng Chen
AU  - Rajula Srivastava
AU  - Niclas Technau
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.02443
UR  - https://arxiv.org/abs/2610.02443
AB  - Let δ∈ [0,1/2] and Q≥ 1. Given a compact C^∞-curve 𝒞 in ℝ^n, denote by N_𝒞(δ, Q) the number of pairs (a, q)∈ ℤ^n× [Q/2,Q] such that the rational point a/q is δ/q-close to 𝒞. We investigate the range of δ in terms of Q which is necessary for the heuristic N_𝒞(δ, Q)asympδ^n−1Q^2 to be correct for the moment curve. For a nondegenerate curve 𝒞⊂ ℝ^n and for sufficiently large Q, Hickman and the second author previously established that N_𝒞(δ, Q)≲_ν δ^n−1Q^2 for all δ∈[Q^α(n+1)+ν,1/2) and any ν>0. Here α(n)=−4n^−2+O(n^−3) was explicitly computed. We show that this range is surprisingly close to being sharp in the asymptotic sense as n→∞, and can at most be extended to δ∈[Q^α(n),1/2). In particular, the sharp exponent can only be about O(n^−3) better, even though it was widely believed before that an improvement of O(n^−1) should be possible. The complementary upper bound in this remaining range has been recently established by Gan--Guo--Oh.
ER  -

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