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How close can rational points get to a manifold?
Abstract
We prove the heuristically predicted lower bound for the number of rational points of height at most Q lying within ε/Q of a fixed analytic nondegenerate manifold in ℝn, provided that εasymp Q−τ for some τ≤ 3/(2m+1), where m is the codimension of the manifold. Our result establishes the lower bound well beyond the previously conjectured range τ≤ 1/m, and improves upon a recent result of Schindler, Srivastava, and Technau, who established the same lower bound for τ≤ 3/(2n−1).
Related
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BibTeX
@misc{beresnevich2026close,
title = {{How close can rational points get to a manifold?}},
author = {Victor Beresnevich and Shreyasi Datta},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06422},
eprint = {2610.06422},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06422},
}
RIS
TY - UNPB
TI - How close can rational points get to a manifold?
AU - Victor Beresnevich
AU - Shreyasi Datta
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06422
UR - https://arxiv.org/abs/2610.06422
AB - We prove the heuristically predicted lower bound for the number of rational points of height at most Q lying within ε/Q of a fixed analytic nondegenerate manifold in ℝ^n, provided that εasymp Q^−τ for some τ≤ 3/(2m+1), where m is the codimension of the manifold. Our result establishes the lower bound well beyond the previously conjectured range τ≤ 1/m, and improves upon a recent result of Schindler, Srivastava, and Technau, who established the same lower bound for τ≤ 3/(2n−1).
ER -