Metrical theorems on fractional parts on real sequences. II
The author continues his study [J. Reine Angew. Math. 396, 192-211 (1989; Zbl 0657.10055)] on fractional parts of real sequences. Let \(a_ n\) be a sequence of reals satisfying (a) \(a_{n+1}-a_ n\geq\lambda\) (or \(\leq -\lambda\)) for some \(\lambda>0\) and for all sufficiently large \(n\); (b) \(| a_ n|>cn^{1+\delta}\) for some \(c,\delta>0\); (c) \(a_ m/a_ n<Km^{-1} n^{-\mu}\) for \(m\leq n^ \sigma\) and for some \(K,\mu,\sigma>0\). The author proves that for almost all real \(x\) there are infinitely many solutions to \[ | a_ n x-m|<(n\log\log(9n))^{-1}\tag{1} \] where \(m\) has no prime factor smaller than \(n^{1+\varepsilon}\) with \(\varepsilon=\min\{\sigma,\delta, \mu\}/4\). Moreover, if \(| a_ n|<n^ A\) for some fixed \(A\), then there are infinitely many solutions to (1) with \(m\) as a prime. The author points out that there are only finitely many solutions for almost all real \(x\) to \[ | a_ n x-m|<n^{-1}(\log\log(9n))^{-1-\eta} \] for any \(\eta>0\) with the above restriction on \(m\). Therefore the above result is very near to best possible. Many useful corollaries can be deduced from the above result. For example, (a) For almost all real \(x\) there are infinitely many solutions to \[ \| a_ n x\|< (n\log\log(9n))^{-1} \] where \(\|\alpha\|=\min\{| \alpha-n|\): \(n\in\mathbb{Z}\}\) for real \(\alpha\). (b) For almost all real \(x\) there are infinitely many fractions \(p/q\) with \[ | x-(p/q)|< \log(q) (q^{3/2}\log\log(q))^{-1} \] where \(p\) is a prime and \(q\) is the square of a prime.
scientific article; zbMATH DE number 205455