Refined shrinking target property of rotations (Q2924834)

The theme of this paper is the inhomogeneous Diophantine approximation, which was initially studied by H. Minkowski more than one century ago. He proved that if \(\theta\) is an irrational number and if \(s\) is real number not of the form \(B\theta - A\) for rational integers \(A\) and \(B\) then the inequality NEWLINE\[NEWLINE \| n \theta - s \| < {1 \over 4 | n |}, NEWLINE\]NEWLINE has infinitely many integer solutions \(n\).NEWLINENEWLINEThe main result of the author is:NEWLINENEWLINE{ Theorem. } Let \(\varphi (n)\) be a monotonically increasing positive function which goes to infinity. For a given irrational \(\theta\) then NEWLINE\[NEWLINE \liminf_{n \to \infty} n \varphi(n) \cdot \| n \theta -s \| = 0 NEWLINE\]NEWLINE for almost every \(s\), if and only if the principal convergent's denominator \(q_k\) of the irrational number \(\theta\) satisfies NEWLINE\[NEWLINE \sum_{k=0}^\infty {\log \{ \min (\varphi(q_k),q_{k+1}/q_k) \} \over \varphi(q_k)}=\infty.NEWLINE\]