Good points for diophantine approximation

Given a sequence (x (n) ) (n=1) (a) of real numbers in the interval [0, 1) and a sequence (delta (n) ) (n=1) (a) of positive numbers tending to zero, we consider the size of the set of numbers in [0, 1] which can be 'well approximated' by terms of the first sequence, namely, those y a [0, 1] for which the inequality |y - x (n) | < delta (n) holds for infinitely many positive integers n. We show that the set of 'well approximable' points by a sequence (x (n) ) (n=1) (a) , which is dense in [0, 1], is 'quite large' no matter how fast the sequence (delta (n) ) (n=1) (a) converges to zero. On the other hand, for any sequence of positive numbers (delta (n) ) (n=1) (a) tending to zero, there is a well distributed sequence (x (n) ) (n=1) (a) in the interval [0, 1] such that the set of 'well approximable' points y is 'quite small'.