Is there a gap between the Hausdorff and the lower Minkowski dimensions?

Question

Does there exist a subset $A\subseteq\mathbb{R}^n$, for some $n$, and numbers $h<m$, such that the Hausdorff dimension $\dim A=h$, while for every cover $A_i$, $A\subseteq\bigcup_{i=1}^\infty A_i$ there exists $i$ such that the lower Minkowski dimension $\underline{\dim}_MA_i>m$?

Answer 1

I believe what you are asking is whether the Hausdorff dimension can be strictly less than the lower packing dimension (also called the lower modified box dimension). I'm pretty sure the strongest formulation, $h=0$ and $m=n,$ is possible, but I don't know of a reference. However, I think examples showing that $h < m$ is possible can be found in Pertti Mattila's 1995 book Geometry of Sets and Measures in Euclidean Spaces and/or in Falconer's 1990 book Fractal Geometry (based on what I wrote in this 13 January 2001 sci.math post), but I don't have either of these books with me now.

The following diagram shows the obvious ordering of the 5 dimensions --- $\dim _{H},$ $\underline{\dim }_{P},$ $\overline{\dim }_{P},$ $\underline{\dim }_{B},$ $\overline{\dim }_{B}$ --- for any fixed choice of a subset of ${\mathbb R}^{n},$ where larger numerical values lie to the right. (I got the diagram by playing around with the LaTeX code from [1], a file that I happened to have on the computer I'm using right now.)

$$\begin{array}{ccccccc} & & & & \overline{\dim }_{P} & & \\ & & & \nearrow & & \searrow & & & & \\ \dim _H & \longrightarrow & \underline{\dim }_{P} & & & & \overline{\dim }_B \\ \ & & & \searrow & \underline{\dim }_B & \nearrow \\ & & & & & & \end{array}$$

What I'm fairly certain is true, but I don't know a reference, is that given any 5 real numbers $\alpha,$ $\beta,$ $\gamma,$ $\delta,$ $\epsilon$ such that

$$0 \;\; \leq \;\; \alpha \;\; \leq \;\; \beta \;\; \leq \;\; \min\{\gamma, \, \delta \} \;\; \leq \;\; \max\{\gamma,\, \delta \} \;\; \leq \;\; \epsilon \;\; \leq \;\; n $$

then there exists a subset $E$ of ${\mathbb R}^n$ such that $\;\dim _{H}(E) = \alpha \;$ and $\;\underline{\dim }_{P}(E) = \beta \;$ and $\;\overline{\dim }_{P}(E) = \gamma \;$ and $\;\underline{\dim }_{B}(E) = \delta \;$ and $\;\overline{\dim }_{B}(E) = \epsilon.$

I thought this was proved in [2] (for the case $n=1),$ but this paper does not include the dimension function $\underline{\dim }_{P}.$ Surely the result can be found somewhere, maybe buried in someone's undergraduate honors thesis or Masters thesis. Searching online, I found several items that discuss the dimension function $\underline{\dim }_{P}$ (such as [3]), but I did not find anything that gives a result like Spear's paper [2] for all 5 of the dimension functions I've brought up.

[1] Dave L. Renfro, A porosity description of the typical continuous graph, Real Analysis Exchange 22 #1 (1996-1997), 70-73.

[2] Donald W. Spear, Sets with different dimensions in $[0,1]$, Real Analysis Exchange 24 #1 (1998-1999), 373-389.

[3] Ying Xiong and Min Wu, Category and dimensions for cut-out sets, Journal of Mathematical Analysis and Applications 358 #1 (1 October 2009), 125-135.