Quantitative non-divergence and lower bounds for points with algebraic coordinates near manifolds
Number Theory
2020-08-18 v2
Abstract
Point counting estimates are a key stepping stone to various results in metric Diophantine approximation. In this paper we use the quantitative non-divergence estimates originally developed by Kleinbock and Margulis to improve lower bounds by Bernik, G\"{o}tze et al. for the number of points with algebraic conjugate coordinates close to a given manifold. In the process, we also improve on a Khinchin-Groshev-type theorem for a problem of constrained approximation by polynomials.
Keywords
Cite
@article{arxiv.2006.10790,
title = {Quantitative non-divergence and lower bounds for points with algebraic coordinates near manifolds},
author = {Alessandro Pezzoni},
journal= {arXiv preprint arXiv:2006.10790},
year = {2020}
}
Comments
Added simplified versions of the main results to the introduction -- removed some superflous hypotheses -- made explicit the last part of the argument in the "Ubiquity" section -- renamed the two main corollaries as theorems -- fixed some typos