PreprintCC0arXiv
Sharp rational points counting near nondegenerate curves in βn
Abstract
We establish sharp bounds for rational points near compact smooth nondegenerate curves in βn, for every nβ₯4. For a nondegenerate curve π in βn, let βπ(Ξ΄,Q) :=#{(p,q)ββ€nΓβ: 1β€ qβ€ Q, dist(p/q,π)β€Ξ΄/q}. We prove that βπ(Ξ΄,Q) β²Ξ΅,π Ξ΄nβ1Q2+QΞ΅ (Q+βk=1nβ1 Ξ΄k2/(k+1)Q(2k+1)/(k+1)). A key new ingredient is a wave packet method developed by Gan--Maldague--Oh.
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BibTeX
@misc{gan2026sharp,
title = {{Sharp rational points counting near nondegenerate curves in $\mathbbR^n$}},
author = {Shengwen Gan and Shaoming Guo and Changkeun Oh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03231},
eprint = {2610.03231},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03231},
note = {CC0},
}
RIS
TY - UNPB
TI - Sharp rational points counting near nondegenerate curves in β^n
AU - Shengwen Gan
AU - Shaoming Guo
AU - Changkeun Oh
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03231
UR - https://arxiv.org/abs/2610.03231
AB - We establish sharp bounds for rational points near compact smooth nondegenerate curves in β^n, for every nβ₯4. For a nondegenerate curve π in β^n, let β_π(Ξ΄,Q) :=#{(p,q)ββ€^nΓβ: 1β€ qβ€ Q, dist(p/q,π)β€Ξ΄/q}. We prove that β_π(Ξ΄,Q) β²_Ξ΅,π Ξ΄^nβ1Q^2+Q^Ξ΅ (Q+β_k=1^nβ1 Ξ΄^k^2/(k+1)Q^(2k+1)/(k+1)). A key new ingredient is a wave packet method developed by Gan--Maldague--Oh.
ER -