Hausdorff dimension of continued fractions with fast growing partial quotients

Let x∈[0,1) be a real number with continued fraction expansion [a1(x),a2(x),a3(x),…], and let pn(x)/qn(x) denote its nth convergent. For a real number α>0 and a function ψ:N→R+ satisfying lim infn→∞ψ(n)>0, we study the sets Eψ(α):=E˜ψ(α):={x∈[0,1):an+1(x)≥qαn(x)ψ(n) for infinitely many n},{x∈[0,1):an+1(x)≥qαn(x)ψ(n) for all sufficiently large n}. We obtain exact formulae for the Hausdorff dimensions of these sets in terms of the exponential and double exponential growth rates of ψ. For Eψ(α), the Hausdorff dimension is determined by the solution of a pressure equation associated with the Gauss map, exhibiting a continuous transition between the classical results of Good (1941) and Sun and Wu (2014). For E˜ψ(α), the Hausdorff dimension depends on the upper double exponential growth rate of ψ and admits a simple explicit expression.