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