A note on metric inhomogeneous Diophantine approximation

Details A note on metric inhomogeneous Diophantine approximation A note on metric inhomogeneous Diophantine approximation

2012-01-22

arxiv.org/abs/1201.4568v1

Mathematics

Number Theory

9 pages

Scientific paper

Suppose that $\varphi(n)$ is a monotone increasing function such that $\sum_n 1/(n\varphi(n))$ diverges. We consider conditions of an irrational $\theta$ for which $$ \liminf_{n \to \infty} n \varphi (n) \| n\theta - s \| = 0 \text{for almost every} s. $$ For a certain class of irrationals the divergence of $\sum_n 1/(n\varphi(n))$ guarantees the limit inferior is 0. But this is not true for general irrational $\theta$. If $\varphi(n)$ increases to infinity, there exist an irrational $\theta$ for which the limit inferior is infinity.

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