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Point counting estimates are a key stepping stone to various results inmetric Diophantine approximation. In this paper we use the quantitativenon-divergence estimates originally developed by Kleinbock and Margulis toimprove lower bounds by Bernik, G\"{o}tze et al. for the number of points withalgebraic conjugate coordinates close to a given manifold. In the process, wealso improve on a Khinchin-Groshev-type theorem for a problem of constrainedapproximation by polynomials.

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