Quantitative non-divergence and lower bounds for points with algebraic coordinates near manifolds

TL;DR

This paper improves lower bounds on the number of algebraic points near manifolds using quantitative non-divergence estimates, advancing results in metric Diophantine approximation and polynomial approximation problems.

Contribution

It applies and enhances Kleinbock and Margulis's non-divergence estimates to obtain sharper bounds for algebraic points near manifolds and improves related approximation theorems.

Findings

Enhanced lower bounds for algebraic points near manifolds

Improved Khinchin-Groshev-type theorem for polynomial approximation

Refined estimates in metric Diophantine approximation

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.