On the growth behavior of partial quotients in continued fractions

Let [a(1)(x), a(2)(x), a(3)(x), ...] be the continued fraction expansion of an irrational number x is an element of (0, 1). It is known that for Lebesgue almost all x is an element of (0, 1) \ Q, (lim inf )(n ->infinity) log a(n)(x)/ log n = 0 and (lim inf )(n ->infinity)log a(n)(x)/ log n = 1. In this note, the Baire classification and Hausdorff dimension of E(alpha, beta) : {x is an element of (0, 1) \ Q:(lim inf )(n ->infinity )log a(n)(x)/ log n=alpha, (lim inf )(n ->infinity) log a(n)(x)/ log n= beta} for all alpha, beta is an element of [0, infinity] with alpha <= beta are studied. We prove that E(alpha, beta) is residual if and only if alpha = 0 and beta = infinity, and the Hausdorff dimension of E(alpha, beta) is as follows: dim(H) E(alpha, beta) = {1, alpha=0; 1/2, alpha > 0.Moreover, the Hausdorff dimension of the intersection of E(alpha, beta) and the set of points with non-decreasing partial quotients is also provided.