A Jarník-type theorem for a problem of approximation by cubic 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 $\left|P(x)\right|\le \psi (H(P))$. 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 $H^g({\mathcal{A}}_n(\psi ))=\infty$ 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 $\psi (r)=r^{-w}$.