The relative growth rate for partial quotients in continued fractions

Abstract

For an irrational number x∈[0,1), let x=[a1(x),a2(x),…] be its continued fraction expansion and {pn(x)qn(x),n≥1} be the sequence of rational convergents. Then, for any α,β∈[0,+∞] with α≤β, the Hausdorff dimension of the following setF(α,β)={x∈[0,1):lim infn→∞log⁡an+1(x)log⁡qn(x)=α,lim supn→∞log⁡an+1(x)log⁡qn(x)=β} admits a dichotomy: it is either 1β+2 or 2β+2 according as α>0 or not.