On the distribution of \(\alpha p\) modulo one
The author proves that if \(\alpha\) is an irrational number then for any real number \(\beta\) there are infinitely many primes \(p\) such that \(\| \alpha p- \beta\|< p^{-4/13}\) where, for any real \(y\), \(\| y\|= \min\{| y-n|\): \(n\in\mathbb{Z}\}\). The result improves on the previous known bound \(p^{-3/10}\) by \textit{G. Harman} [J. Lond. Math. Soc., II. Ser. 27, 9-18 (1983; Zbl 0504.10018)]. In the proof the author applies sieve methods with more delicate techniques and estimates than the previous authors.
scientific article; zbMATH DE number 483986