On the exceptional sets concerning the leading partial quotient in continued fractions

Abstract

This paper is concerned with the fractal structure of the largest partial quotient in continued fractions. Let [a1(x),a2(x),⋯] be the continued fraction expansion of x∈(0,1). Denote by Tn(x) the largest digit among the first n partial quotients of x. For any real numbers 0<α<β<∞, one is interested in the Hausdorff dimension of the exceptional setFϕ(α,β)={x∈(0,1):lim infn→∞Tn(x)ϕ(n)=α,lim supn→∞Tn(x)ϕ(n)=β} when ϕ(n) tends to infinity with polynomial or exponential rates. It is shown that the Hausdorff dimension of Fϕ(α,β) decreases as the increasing speed of ϕ(n) rises. As a result, we supplement the results got by Wu and Xu, Liao and Rams. In a similar way, the corresponding exceptional sets for the sums of partial quotients are also studied.