APPROXIMATION OF IRRATIONAL NUMBERS BY PAIRS OF INTEGERS FROM A LARGE SET

We show that there is a set $S \subseteq {\mathbb N}$ with lower density arbitrarily close to $1$ such that, for each sufficiently large real number $\alpha $ , the inequality $|m\alpha -n| \geq 1$ holds for every pair $(m,n) \in S^2$ . On the other hand, if $S \subseteq {\mathbb N}$ has density $1$ , then, for each irrational $\alpha>0$ and any positive $\varepsilon $ , there exist $m,n \in S$ for which $|m\alpha -n|<\varepsilon $ .