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

Andrew Mitchell; Lars Olsen

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

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arxiv 1812.09542 v1 pith:DTDK67AX submitted 2018-12-22 math.MG

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

Andrew Mitchell , Lars Olsen This is my paper

classification math.MG

keywords dimensionequalcompacthausdorfflowerpackingassouadcoincidence

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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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Intersections of continuous functions with prescribed Hausdorff dimension s and box dimensions r,t are shown to be strongly c-algebrable and spaceable, plus related lineability results.