论文标题
在基础上有两个扩展的某些集合的尺寸上
On the dimension of certain sets araising in the base two expansion
论文作者
论文摘要
我们表明，对于基础，两个扩展\ [x = \ sum_ {i = 1}^{\ infty} 2^{ - （d_ {1}（x）（x）+d_ {2}（x）+dots+dots+dots+dots+dots+dots+dots+d_ {i}（x）（x）} \] $d_{i}(x)\in\mathbb{N}$ the set $A=\{x|\lim_{i\to\infty}d_{i}(x)=\infty\}$ has Hausdorff dimension zero, this is opposed to a result on the continued fraction expansion, here $A$ has Hausdorff dimension $1/2$, see \ cite {[go]}。
We show that for the base two expansion \[ x=\sum_{i=1}^{\infty}2^{-(d_{1}(x)+d_{2}(x)+\dots+d_{i}(x))}\] with $x\in(0,1]$ and $d_{i}(x)\in\mathbb{N}$ the set $A=\{x|\lim_{i\to\infty}d_{i}(x)=\infty\}$ has Hausdorff dimension zero, this is opposed to a result on the continued fraction expansion, here $A$ has Hausdorff dimension $1/2$, see \cite{[GO]}. Furthermore we construct subsets of $B=\{x|\limsup_{i\to\infty}d_{i}(x)=\infty\}$ which have Hausdorff dimension one and find a dimension spectrum in set $B$.
