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Inhomogeneous Diophantine approximation on integer polynomials with non-monotonic error function. (English) Zbl 1292.11086
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 \[ | P(x) - d | < \psi(H(P)), \] 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 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.

11J83 Metric theory
11K60 Diophantine approximation in probabilistic number theory
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