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

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, Götze 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.

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

References in corpus (2)

• Ubiquity and a general logarithm law for geodesics
• Systems of small linear forms and Diophantine approximation on manifolds