Metric diophantine approximation and probability
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Let pn=qn = (pn=qn)(x) denote the nth simple continued fraction convergent to an arbitrary irrational number x (0; 1). Define the sequence of approximation constants n(x) := qn2|x - p n/qn|. It was conjectured by Lenstra that for almost all x (0; 1), lim n1/n|{j : 1 j n and j (x) z}| = F(z) where F(z) := z/log 2 if 0 z 1/2, and 1/log 2 (1-z+log(2z)) if 1/2 z 1. This was proved in [BJW83] and extended in [Nai98] 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.
