Let \(\theta \) be an irrational number and \(\varphi a real number. \(C > 2\) \(\varepsilon 0\). There are infinitely many positive integers n free of prime factors \(> (\log )^C\), such that \(\Vert \theta + \varphi \Vert < n^{-\left( \frac{1}{3} - \frac{2}{3C}\right) \varepsilon }\). Here y\Vert is the distance from y to \(\mathbb Z\).