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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.

##### MSC:
 11J83 Metric theory 11K60 Diophantine approximation in probabilistic number theory
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