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arXiv · 2201.09641

On the dimension of certain sets araising in the base two expansion

Jörg Neunhäuserer
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Abstract

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 $B$.

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Jörg Neunhäuserer. 2022-01-24. On the dimension of certain sets araising in the base two expansion. https://arxiv.org/abs/2201.09641

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