A Jarn\'ik-type theorem for a problem of approximation by cubic polynomials

TL;DR

This paper extends Jarník-type theorems to polynomial approximation, establishing Hausdorff measure results for sets of real numbers approximable by cubic polynomials with bounded discriminant.

Contribution

It proves the convergence case of Hausdorff measure for polynomial approximation with bounded discriminant, completing the case for degree three and specific functions.

Findings

Hausdorff measure of approximation sets is infinite under divergence conditions

Complete solution for degree three polynomial approximation with specific decay functions

Extension of classical approximation theorems to bounded discriminant polynomials

Abstract

For a given decreasing positive real function $\psi$, let $\mathcal{A}_n(\psi)$ be the set of real numbers for which there are infinitely many integer polynomials $P$ of degree up to $n$ such that $\left\lvert P(x) \right\rvert \leq \psi(\operatorname{H}(P))$. A theorem by Bernik states that $\mathcal{A}_n(\psi)$ has Hausdorff dimension $\frac{n+1}{w+1}$ in the special case $\psi(r) = r^{-w}$, while a theorem by Beresnevich, Dickinson and Velani implies that the Hausdorff measure $\operatorname{\mathcal{H}}^g(\mathcal{A}_n(\psi))=\infty$ 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 $\psi(r) = r^{-w}$.