Let ε ： N →4 R be a parameter function satisfying the condition ε （k） ＋ k ＋ 1 〉 0 and let Tε ： （0, 1] → （0, 1] be a transformation defined by Tε（x）=[-1＋（k＋1）x]/[1＋k-kεx] for x∈ （ 1/（k＋1）,...Let ε ： N →4 R be a parameter function satisfying the condition ε （k） ＋ k ＋ 1 〉 0 and let Tε ： （0, 1] → （0, 1] be a transformation defined by Tε（x）=[-1＋（k＋1）x]/[1＋k-kεx] for x∈ （ 1/（k＋1）, 1/k]. Under the algorithm Tε, every x ∈ （0, 1] is attached an expansion, called generalized continued fraction （GCFε） expansion with parameters by Schweiger. Define the sequence {kn （x）}n≥l of the partial quotients of x by k1 （x） =「1/x」 and kn （x） = k1 （T^（n-1）ε （x）） for every n≥ 2. Under the restriction -k - 1 〈 ε（k） 〈 -k, define the set of non-recurring GCFε expansions asFε= {x ∈（0, 1] ： kn＋1（x） 〉 kn（x） for infinitely many n}. It has been proved by Schweiger that Fε has Lebesgue measure 0. In the present paper, we strengthen this result by showing that {dimH Fε≥1/2 ,when ε（k）=-k-1＋ρ for a constant 0 〈 ρ 〈 1;1/（s＋2）≤dimH Fε ≤1/s,when ε（k）=-k-1＋1/k^s for any s≥1 where dimH denotes the Hausdorff dimension.展开更多
基金Supported by the National Natural Science Foundation of China(Grant No.11361025)
文摘Let ε ： N →4 R be a parameter function satisfying the condition ε （k） ＋ k ＋ 1 〉 0 and let Tε ： （0, 1] → （0, 1] be a transformation defined by Tε（x）=[-1＋（k＋1）x]/[1＋k-kεx] for x∈ （ 1/（k＋1）, 1/k]. Under the algorithm Tε, every x ∈ （0, 1] is attached an expansion, called generalized continued fraction （GCFε） expansion with parameters by Schweiger. Define the sequence {kn （x）}n≥l of the partial quotients of x by k1 （x） =「1/x」 and kn （x） = k1 （T^（n-1）ε （x）） for every n≥ 2. Under the restriction -k - 1 〈 ε（k） 〈 -k, define the set of non-recurring GCFε expansions asFε= {x ∈（0, 1] ： kn＋1（x） 〉 kn（x） for infinitely many n}. It has been proved by Schweiger that Fε has Lebesgue measure 0. In the present paper, we strengthen this result by showing that {dimH Fε≥1/2 ,when ε（k）=-k-1＋ρ for a constant 0 〈 ρ 〈 1;1/（s＋2）≤dimH Fε ≤1/s,when ε（k）=-k-1＋1/k^s for any s≥1 where dimH denotes the Hausdorff dimension.
