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 $\psi :\mathbb{N}\to (0,1/2)$ with ${\sum }_q\psi (q)=\infty$ and any number γ, the following set$W(\psi ,\gamma )=\{x\in [0,1]:|qx-p-\gamma |<\psi (q)\text{ for infinitely many }q,p\in \mathbb{N}\}$ has full Lebesgue measure. Since then, there are very few results in relaxing the monotonicity condition. In this paper, we show that if γ is can not be approximate by rational numbers too well, then the monotonicity condition can be replaced by the upper bound condition$\psi (q)=O({(q{(\log \log q)}^2)}^{-1}).$ In particular, this covers the case when γ is not Liouville, for example $\pi ,e,\ln 2,\sqrt{2}$. In general, if γ is irrational, $\psi (q)=O(q^{-1}{(\log \log q)}^{-2})$ and in addition,$(\underset{Q\to \infty }{\lim \inf }\sum _{q=Q}^{Q^{{(\log Q)}^{1/8}}}\psi (q))=\infty ,$ then $W(\psi ,\gamma )$ has full Lebesgue measure. Our proof is based on a quantitative study of the discrepancy for irrational rotations.

1958年，Szüsz证明了Khintchine关于Diophantine逼近定理的不均匀版本。舒兹定理指出，对于任何非递增的逼近函数$\psi ：\mathbb{ñ}\to （0，\mathrm{1个}/\mathrm{2个}）$ 和 ${\sum }_q\psi （q）=\infty$和任意数γ，以下集合$\mathrm{w\; ^}（\psi ，\gamma ）=\{X\in [0，\mathrm{1个}]：|qX-p-\gamma |<\psi （q）\text{ 无限地 }q，p\in \mathbb{ñ}\}$具有完整的Lebesgue量度。从那时起，在放松单调性条件方面几乎没有结果。在本文中，我们表明，如果不能很好地用有理数逼近γ，则可以用上限条件代替单调性条件$\psi （q）=Ø（{（q{（\mathrm{日志}\mathrm{日志}q）}^{\mathrm{2个}}）}^{-\mathrm{1个}}）。$特别地，这涵盖了例如当γ不是Liouville时的情况。$\pi ，Ë，\ln \mathrm{2个}，\sqrt{\mathrm{2个}}$。通常，如果γ不合理，$\psi （q）=Ø（q^{-\mathrm{1个}}{（\mathrm{日志}\mathrm{日志}q）}^{-\mathrm{2个}}）$ 另外，$（\underset{问\to \infty }{\mathrm{林}\mathrm{信息}}\sum _{q=问}^{{问}^{{（\mathrm{日志}问）}^{\mathrm{1个}/8}}}\psi （q））=\infty ，$ 然后 $\mathrm{w\; ^}（\psi ，\gamma ）$具有完整的Lebesgue量度。我们的证明是基于对非理性旋转差异的定量研究。