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arXiv · 1812.09542

Coincidence and noncoincidence of dimensions in compact subsets of $[0,1]$

Andrew Mitchell·Lars Olsen
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Abstract

We show that given any six numbers $r,s,t,u,v,w \in (0,1]$ satisfying $r \leq s \leq \min(t,u) \leq \max(t,u) \leq v \leq w$, it is possible to construct a compact subset of $[0,1]$ with Hausdorff dimension equal to $r$, lower modified box dimension equal to $s$, packing dimension equal to $t$, lower box dimension equal to $u$, upper box dimension equal to $v$ and Assouad dimension equal to $w$. Moreover, the set constructed is an $r$-Hausdorff set and a $t$-packing set.

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Andrew Mitchell, Lars Olsen. 2018-12-22. Coincidence and noncoincidence of dimensions in compact subsets of $[0,1]$. https://arxiv.org/abs/1812.09542

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