Alessandro Pezzoni | Searcharxiv

arXiv2020

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

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.

math.NT↗

arXiv2018

A Jarník-type theorem for a problem of approximation by cubic polynomials

For a given decreasing positive real function $ψ$, let $\mathcal{A}_n(ψ)$ be the set of real numbers for which there are infinitely many integer polynomials $P$ of degree up to $n$ such that $\left\lvert P(x) \right\rvert \leq ψ(\operatorname{H}(P))$. A theorem by Bernik states that $\mathcal{A}_n(ψ)$ has Hausdorff dimension $\frac{n+1}{w+1}$ in the special case $ψ(r) = r^{-w}$, while a theorem by Beresnevich, Dickinson and Velani implies that the Hausdorff measure $\operatorname{\mathcal{H}}^g(\mathcal{A}_n(ψ))=\infty$ when a certain series diverges. In this paper we prove the convergence counterpart of this result when $P$ has bounded discriminant, which leads to a complete solution when $n = 3$ and $ψ(r) = r^{-w}$.

math.NT↗