Diophantine approximation with square-free numbers (Q793772)

Let \(\alpha\) be a given real irrational number, and let \(\theta<2/3\). It is shown that there exist infinitely many square-free integers m,n such that \(| n\alpha -m| \leq n^{-\theta}.\) This was previously known only for \(\theta<1/2\) [\textit{G. Harman} (see the preceding review)]. The method, which is elementary, is quite different from Harman's. It is based on a result from the geometry of numbers, together with the author's ''square sieve'' [Math. Ann. 266, 251-259 (1984; Zbl 0514.10038)].