Two problems in metric diophantine approximation, II

AbstractFor an arbitrary sequence {αn} of nonnegative real numbers there is no known necessary and sufficient condition that for almost all x (in the sense of Lebesgue measure) there are infinitely many fractions pq satisfying |x − pq| < αqq. With a restriction on {αn} weaker than any previously used, except in a recent result of Erdös, we solve this problem and the analogous problem where p and q are required to be relatively prime