On the metric theory of inhomogeneous Diophantine approximation: An Erdős-Vaaler type result

Abstract

In 1958, Szüsz proved an inhomogeneous version of Khintchine's theorem on Diophantine approximation. Szüsz's theorem states that for any non-increasing approximation function $\ddot{I}ˆ:\mathbb{N}\to (0,1/2)$ with ${\sum }_q\ddot{I}ˆ(q)=\infty$ and any number ${\hat{I}}^3,$ the following set \[ W(ψ,γ)=\{x\in [0,1]: |qx-p-γ|< ψ(q) \text{ for infinitely many } q,p\in\mathbb{N}\} \] has full Lebesgue measure. Since then, there are very few results in relaxing the monotonicity condition. In this paper, we show that if ${\hat{I}}^3$ is can not be approximate by rational numbers too well, then the monotonicity condition can be replaced by the upper bound condition $\ddot{I}ˆ(q)=O((q(\log \log q{)}^2{)}^{-1}).$ In particular, this covers the case when ${\hat{I}}^3$ is not Liouville, for example $\ddot{I}€,e,\ln 2,\sqrt{2}.$ In general, if ${\hat{I}}^3$ is irrational, $\ddot{I}ˆ(q)=O(q^{-1}(\log \log q{)}^{-2})$ and in addition, \[ \left(\liminf_{Q\to\infty} \sum_{q=Q}^{Q^{(\log Q)^{1/8} }}ψ(q)\right)=\infty, \] then $W(\ddot{I}ˆ,{\hat{I}}^3)$ has full Lebesgue measure. Our proof is based on a quantitative study of the discrepancy for irrational rotations.

extended some of the results in an earlier version