Hausdorff dimension of certain sets arising in continued fraction expansions

This paper is concerned with the fractional dimensions of some sets of points with their partial quotients obeying some restrictions in their continued fraction expansions. The Hausdorff dimension of the following set, which shares a dichotomy law according to Borel–Bernstein's theorem, is completely determinedE(ϕ)={x∈[0,1):an(x)⩾ϕ(n), i.o. n}. Our result significantly strengthens the results of Good and Lúczak.