Increasing rate of weighted product of partial quotients in continued fractions

Abstract

Let [a1(x),a2(x),…,an(x),…] be the continued fraction expansion of x∈[0,1). In this paper, we study the increasing rate of the weighted product ant0(x)an+1t1(x)⋯an+mtm(x) ,where ti∈R+(0≤i≤m) are weights. More precisely, let φ:N→R+ be a function with φ(n)/n→∞ as n→∞. For any (t0,…,tm)∈R+m+1 with ti≥0 and at least one ti≠0(0≤i≤m), the Hausdorff dimension of the set E̲({ti}i=0m,φ)=x∈[0,1):lim infn→∞logant0(x)an+1t1(x)⋯an+mtm(x)φ(n)=1 is obtained. Under the condition that (t0,…,tm)∈R+m+1 with 0<t0≤t1≤⋯≤tm, we also obtain the Hausdorff dimension of the set E¯({ti}i=0m,φ)=x∈[0,1):lim supn→∞logant0(x)an+1t1(x)⋯an+mtm(x)φ(n)=1.