Inhomogeneous approximation by coprime integers (Q416794)

The inhomogeneous version of the Dirichlet's theorem proved by Minkowski is: for any irrational \(\alpha\in \mathbb R\) and for any \(\gamma\in \mathbb R\backslash (\alpha\mathbb Z+\mathbb Z)\) there are infinitely many \(m,n\in\mathbb Z\) for which NEWLINE\[NEWLINE|n\alpha-m-\gamma|\leq\frac{1}{4|n|}.NEWLINE\]NEWLINE The author addresses the problem of obtaining analogous results with \(m\) and \(n\) coprime. The best known result for this problem was a recent result of \textit{M. Laurent} and \textit{A. Nogueira} [Acta Arith. 154, No. 4, 413-427 (2012; Zbl 1273.11106)]: For any irrational \(\alpha\in\mathbb R\) and for any \(\gamma\in \mathbb R\) there are infinitely many pairs of coprime integers \((m,n)\) such that NEWLINE\[NEWLINE |n\alpha-m-\gamma|\leq \frac{c}{|n|^{1/2}} \tag{e1}NEWLINE\]NEWLINE where \(c\) is a constant depending only on \(\alpha\) and \(\gamma\).NEWLINENEWLINEWith a method using \textit{elementary techniques} (relying on classical results on fractions and estimates from number theory) different from the non elementary proofs of Laurent-Nogueira (loc. cit.) (relying on estimates for the density of orbits of points in \(\mathbb R^2\) under the action of \(\mathrm{SL}_2(\mathbb Z)\)\big), they improve Laurent-Nogueira (loc. cit.) by the following results:NEWLINENEWLINETheorem. Let \(c>2\sqrt{\log 2}\). For any irrational \(\alpha\in \mathbb R\) and for any \(\gamma\in\mathbb R\) there are infinitely many pairs of coprime integers \((m,n)\) such that NEWLINE\[NEWLINE|n\alpha-m-\gamma|\leq\frac{\exp(c\sqrt{\log |n|)}}{|n|}.NEWLINE\]NEWLINE As \(n\to\infty\) the function \(\exp(c\sqrt{\log n})\) grows asymptotically more slowly than any power of \(n\), which implies the following corollary:NEWLINENEWLINEFor any irrational \(\alpha\in \mathbb R\) and for any \(\gamma\in \mathbb R\) and \(\varepsilon>0\) there are infinitely many pairs of coprime integers \((m,n)\) such that NEWLINE\[NEWLINE |n\alpha-m-\gamma|\leq\frac{1}{|n|^{1-\varepsilon}}, \tag{e2}NEWLINE\]NEWLINE approximation clearly better than ({e1}).