参数函数$\epsilon$取边际值时的$GCF_\epsilon$集

Let ∈ :N → R be a parameter function satisfying the condition ∈(k) + k + 1 > 0and let T∈ :(0,1] →(0,1] be a transformation defined by T∈(x) =-1 +(k + 1)x1 + k-k∈x for x ∈(1k + 1,1k].Under the algorithm T∈,every x ∈(0,1] is attached an expansion,called generalized continued fraction(GCF∈) expansion with parameters by Schweiger.Define the sequence {kn(x)}n≥1of the partial quotients of x by k1(x) = ∈1/x∈ and kn(x) = k1(Tn-1∈(x)) for every n ≥ 2.Under the restriction-k-1 < ∈(k) <-k,define the set of non-recurring GCF∈expansions as F∈= {x ∈(0,1] :kn+1(x) > kn(x) for infinitely many n}.It has been proved by Schweiger that F∈has Lebesgue measure 0.In the present paper,we strengthen this result by showing that{dim H F∈≥12,when ∈(k) =-k-1 + ρ for a constant 0 < ρ < 1;1s+2≤ dimHF∈≤1s,when ∈(k) =-k-1 +1ksfor any s ≥ 1where dim H denotes the Hausdorff dimension.

Some Sets of GCF$_\epsilon$ Expansions Whose Parameter $\epsilon$ Fetch the Marginal Value

Let $\epsilon: \mathbb{N}\to \mathbb{R}$ be a parameter function satisfying the condition $\epsilon(k)+k+1> 0$ and let $T_{\epsilon}:(0,1]\to (0,1]$ be a transformation defined by $$T_{\epsilon}(x)=\frac{-1+(k+1)x}{1+k-k\epsilon x} \ {\text{for}}\ x\in \Big(\frac{1}{k+1},\frac{1}{k}\Big].$$ Under the algorithm $T_{\epsilon}$, every $x\in (0,1]$ is attached an expansion, called generalized continued fraction (GCF$_{\epsilon}$) expansion with parameters by Schweiger. Define the sequence $\{k_n(x)\}_{n\ge 1}$ of the partial quotients of $x$ by $k_1(x)=\lfloor 1/x\rfloor$ and $k_n(x)=k_1(T_{\epsilon}^{n-1}(x))$ for every $n\ge 2$. Under the restriction $-k-1<\epsilon(k)<-k$, define the set of non-recurring GCF$_{\epsilon}$ expansions as $$\mathcal{F}_{\epsilon}=\{x\in (0,1]: k_{n+1}(x)>k_n(x)\ {\text{for infinitely many }}\ n\}.$$ It has been proved by Schweiger that $ \mathcal{F}_{\epsilon}$ has Lebesgue measure 0. In the present paper, we strengthen this result by showing that \begin{eqnarray*} \left\{\begin{array}{ll}\dim_H \mathcal{F}_{\epsilon}\ge \frac{1}{2}, & \text{when $\epsilon(k)=-k-1+\rho$ for a constant $0<\rho<1$;} \frac{1}{s+2}\le\dim_H \mathcal{F}_{\epsilon}\le \frac{1}{s}, & \text{when $\epsilon(k)=-k-1+\frac{1}{k^s}$ for any $s\ge1$}\end{array}\right.\end{eqnarray*} where $\dim_H$ denotes the Hausdorff dimension.