Irrationality exponent and convergence exponent in continued fraction expansions

Let x is an element of (0, 1) be an irrational number with continued fraction expansion [a(1)(x), a(2)(x), . . . , a(n)(x), . . .]. We give the multifractal spectrum of the irrationality exponent and the convergence exponent of x defined by nu (x) := sup {nu > 0 : |x- p/q| < 1/q(nu) for infinitely many (q, p) is an element of N x Z} and tau (x) := inf {s >= 0 : Sigma(n >= 1)a(n)(-s) (x) < infinity} respectively. To be precise, we completely determine the Hausdorff dimension of E (alpha, nu) = {x is an element of (0, 1) : tau(x) = alpha, nu(x) = nu} for any alpha >= 0, nu >= 2.