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

Alessandro Pezzoni¹

¹ Department of Mathematics University of York York, YO10 5DD, United Kingdom

Acta Arithmetica, Tome 193 (2020) no. 3, pp. 269-281

Résumé

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 $\psi (r) = r^{-w}$.

Détail

DOI : 10.4064/aa180926-9-5

Keywords: given decreasing positive real function psi mathcal psi set real numbers which there infinitely many integer polynomials degree leq psi operatorname theorem bernik states mathcal psi has hausdorff dimension frac special psi w while theorem beresnevich dickinson velani implies hausdorff measure satisfies mathcal mathcal psi infty certain series diverges paper prove convergence counterpart result has bounded discriminant which leads complete solution psi w