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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 $| P(x) |\leq \psi ( \operatorname{H}(P))$. 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 satisfies $\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 $\psi (r) = r^{-w}$.