For a given decreasing positive real function ψ, let An(ψ) be the set of real numbers for which there are infinitely many integer polynomials P of degree up to n such that |P (x)| ≤ ψ(H(P )). A theorem by Bernik states that An(ψ) has Hausdorff dimension 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 H g (An(ψ)) = ∞ 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 .
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.