A ``zero-one'' law for diophantine approximations (Q1381650)

\textit{P. X. Gallagher} proved that the set of real \(\alpha\) for which the inequality \(|\alpha-{m\over n}|<{\delta(n)\over n}\), where \(\delta (n)\geq 0\) and \((m,n)=1\), holds for infinitely many rationals \(m/n\), or its complement has zero Lebesgue measure [J. Math. Soc. Japan 13, 342-345 (1962; Zbl 0106.04106)]. Such ``zero-one'' laws arise frequently in metric diophantine approximation [see \textit{G. Harman}'s Metric number theory, Oxford University Press (1998)] reflecting the natural connection with probability. The author shows that restricting the rationals suitably can destroy the validity of a zero-one law but also shows that one holds when the approximating rationals \(m/n\) lie in a set \(V\) which is invariant with respect to the transformations \(T_p\), \(T_q\) \((T_p(V) =T_q(V)=V)\), where \(p,q\) are primes, \(T_u(x) =ux\), \(\delta(n)\geq 0\), \(\delta (n)=o(n)\) and \(\delta(pn) \asymp\delta (qn)\asymp \delta(n)\).