Approximation of Irrational Numbers by Pairs of Integers From a Large Set

We show that there is a set S ⊆ N with lower density arbitrarily close to $1$ such that, for each sufficiently large real number α , the inequality |mα -n| ≥ 1 holds for every pair (m,n) ∈ S² . On the other hand, if S ⊆ N has density $1$ , then, for each irrational α>0 and any positive ε , there exist m,n ∈ S for which |mα -n|<ε .