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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

Andrew Mitchell , Lars Olsen This is my paper · ORCID
classification math.MG
keywords dimensionequalcompacthausdorfflowerpackingassouadcoincidence
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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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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. The sequence property for fractal dimensions

    math.MG 2026-08 accept novelty 7.0 of 10

    Finitely stable fractal dimensions of compact sets are always matched by some convergent sequence inside the set, and dimension-homogeneous sets admit one sequence matching many dimensions simultaneously.

  2. Assouad dimension of the Takagi function

    math.CA 2025-02 conditional novelty 7.0 of 10

    The Assouad dimension of the graph of every generalized Takagi function with limsup b^n|c_n| < ∞ is 1, and for T_{a,b} this happens exactly when a ≤ 1/b.

  3. On strong algebrability and spaceability of continuous functions and fractal dimensions

    math.FA 2026-06 unverdicted novelty 5.0 of 10

    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.

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