Inhomogeneous Diophantine approximation on integer polynomials with non-monotonic error function (Q2847830)

Let \(n \in {\mathbb N}\), \(n \geq 2\), let \(d \in {\mathbb R}\) and let \(\psi: {\mathbb R}_+ \rightarrow {\mathbb R}_+\) be some function. The authors consider the set of real numbers \(x\) for which there are infinitely many integer polynomials \(P\) with \(\deg P \leq n\) such that NEWLINE\[NEWLINE | P(x) - d | < \psi(H(P)), NEWLINE\]NEWLINE where \(H(P)\) denotes the maximum absolute value among the coefficients of \(P\). It is shown that is the series \(\sum_{h=1}^\infty h^{n-1} \psi(h)\) converges, then this set is a null set with respect to the Lebesgue measure. This strengthens and extends a result of \textit{V. I. Bernik} [Acta Arith. 53, No. 1, 17--28 (1989; Zbl 0692.10042)], who proved this to be true for \(d \in {\mathbb Q}\) on the additional assumption that \(\psi\) is monotonic. The latter assumption is removed in the paper under review.