Sylvester连分数展式中若干例外集的Hausdorff维数OACSTPCD

For x ∈(0,1],let x=[d1,d2,…]be its Sylvester continued fraction expansions,we calculate the Hausdorff dimension of the set A(ψ)defined in terms of the Sylvester continued fraction expansions as A(ψ)={x∈(0,1]:limn→∞ logdn(x)/ψ(n)=1},where ψ(n)is a positive function defined on N.By constructing the covering and a suitable subset of Cantor,we get the exact Hausdorff dimension of the set as dim H A(ψ)=lim infn→∞ ψ(1)+ψ(2)+…+ψ(n)/ψ(n+1)At the same time,we also calculate the Hausdorff dimension of the set of points with limn→∞logdn(x)/ψ(n)=0 or ∞.