On Kurzweil's 0-1 law in inhomogeneous Diophantine approximation (Q2804251)

The paper under review is concerned with the study of a particular problem in metric Diophantine approximation. The fundamental question in this study is how well can a real number be approximated by rationals. To state the exact problem, some notation is needed. Let \(\psi(n)\) be a positive, non-increasing sequence. Let \(\theta\in\mathbb R\) be fixed, then consider the problem of finding the necessary and sufficient conditions for the inequalityNEWLINENEWLINENEWLINE\[NEWLINE\left\| n\theta-s\right\|<\psi(n)\quad \text{for \;infinitely\;many \;} n\in\mathbb NNEWLINE\]NEWLINENEWLINENEWLINEto hold for almost all \(s\in\mathbb R\). Here, \(\| \, \|\) denotes the distance to the nearest integer. The main result of the paper is as follows.NEWLINENEWLINETheorem. Let \(\psi(n)\) be as above and \(\theta\) be an irrational number with principle convergents \(p_k/q_k\). Then, for almost all \(s\in\mathbb R\),NEWLINENEWLINENEWLINE\[NEWLINE\left\| n\theta-s\right\|<\psi(n)\quad \text{for \;infinitely\;many \;} n\in\mathbb NNEWLINE\]NEWLINE if and only if NEWLINE\[NEWLINE \sum_{k=0}^\infty\left(\sum_{n=q_k}^{q_{k+1}-1}\min(\psi(n), \|q_k\theta\|)\right)=\infty.NEWLINE\]NEWLINENEWLINENEWLINEThe proof of this theorem naturally reduces to two parts, convergence and divergence. For the convergence part the proof relies on natural covering arguments, which when equipped together with the first Borel-Cantelli lemma prove the desired result. The divergence case is more intricate and relies upon a careful analysis of the size of the sequences, which together with the second Borel-Cantelli lemma yield the result.NEWLINENEWLINEIn the paper, a couple of important consequences of the main theorem are also discussed. These include some theorems of \textit{J. Tseng} [Discrete Contin. Dyn. Syst. 20, No. 4, 1111--1122 (2008; Zbl 1151.37004)] and \textit{J. Kurzweil} [Stud. Math. 15, 84--112 (1955; Zbl 0066.03702)], and that of Khintchine sequences.NEWLINENEWLINEAt the end of the paper, an analogue of the main theorem over the field of formal Laurent series is also proven.