Metric Diophantine Approximation and Probability

Let pn qn pn qn x denote the nth simple continued fraction convergent to an arbitrary irrational number x De ne the sequence of approximation constants n x q njx pn qnj It was conjectured by Lenstra that for almost all x lim n n jfj j n and j x zgj F z where F z z log if z and log z log z if z This was proved in BJW and extended in Nai to the same conclusion for kj x where kj is a sequence of positive integers satisfying a certain technical condition related to ergodic theory Our main result is that this condition can be dispensed with we only need that kj be strictly increasing