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

In 1958, Szüsz proved an inhomogeneous version of Khintchine's theorem on Diophantine approximation. Szüsz's states that for any non-increasing approximation function ψ:N→(0,1/2) with ∑qψ(q)=∞ and number γ, the following setW(ψ,γ)={x∈[0,1]:|qx−p−γ|<ψ(q) infinitely many q,p∈N} has full Lebesgue measure. Since then, there are very few results in relaxing monotonicity condition. this paper, we show if γ is can not be approximate by rational numbers too well, then condition replaced upper bound conditionψ(q)=O((q(log⁡log⁡q)2)−1). particular, covers case when Liouville, example π,e,ln⁡2,2. general, irrational, ψ(q)=O(q−1(log⁡log⁡q)−2) addition,(liminfQ→∞∑q=QQ(log⁡Q)1/8ψ(q))=∞, W(ψ,γ) Our proof based a quantitative study discrepancy irrational rotations.