Order of magnitude of the partial quotients of regular continued fraction expansion

For any irrational x is an element of [0, 1], let [a(1)(x), a(2)(x), ...] d enote the regular continued fraction expansion of x and define f, for all z > 0 by f(z) := exp(-1/z log (2)) and f(n) by [GRAPHICS] J. GALAMBOS proved that (mu the Gauss measure) [GRAPHICS] In this paper, we first point out that for all z > 0, (f(n)(x,z))(n) has no limit for n --> infinity for almost all x is an element of [0, 1], proving more precisely that: For all z > 0, one has for almost all x is an element of [0, 1] [GRAPHICS] Then we prove mainly the more precise result: For all z > 0, the sequence ( f(n))(n) has no subsequence (f(nk))(k) which converges almost everywhere.