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 ψ:N→(0,1/2) with ∑qψ(q)=∞ and any number γ, the following setW(ψ,γ)={x∈[0,1]:|qx−p−γ|<ψ(q) for infinitely many q,p∈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 conditionψ(q)=O((q(log⁡log⁡q)2)−1). In particular, this covers the case when γ is not Liouville, for example π,e,ln⁡2,2. In general, if γ is irrational, ψ(q)=O(q−1(log⁡log⁡q)−2) and in addition,(liminfQ→∞∑q=QQ(log⁡Q)1/8ψ(q))=∞, then W(ψ,γ) has full Lebesgue measure. Our proof is based on a quantitative study of the discrepancy for irrational rotations.