Hausdorff dimension of sets with restricted, slowly growing partial quotients

I. J. Good [Proc. Cambridge Philos. Soc. 37 (1941), pp. 199–228] showed that the set of irrational numbers in (0,1) whose partial quotients a_n tend to infinity is of Hausdorff dimension 1/2. A number of related results impose restrictions of the type a_n\in B or a_n\geq f(n), where B is an infinite subset of \mathbb N and f is a rapidly growing function with n. We show that, for an arbitrary B and an arbitrary f with values in [\min B,\infty ) and tending to infinity, the set of irrational numbers in (0,1) such that \[ a_n\in B,\ a_n\leq f(n)\text { for all n\in \mathbb N, and }a_n\to \infty \text { as }n\to \infty \] is of Hausdorff dimension \tau (B)/2, where \tau (B) is the exponent of convergence of B.