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 $\underline{\dim}_MA_i>m$?

1$\begingroup$ I believe what you're asking about is whether the Hausdorff dimension can be strictly less than the lower packing dimension (also called the lower modified box dimension). I'd be surprised if $h=0$ and $m=n$ is not possible. I don't have these books with me now to know for sure, but I think examples might be found in Pertti Mattila's 1995 book Geometry of Sets and Measures in Euclidean Spaces or in Falconer's 1990 book Fractal Geometry (based on what I wrote in this 13 January 2001 sci.math post). $\endgroup$– Dave L RenfroMar 28 '17 at 21:01

$\begingroup$ @DaveLRenfro Thanks, I forgot about the lower packing! Actually Mattila sends the reader to Tricot "Two definitions of fractional dimension" (1982). Please write this as an answer and I will accept it. $\endgroup$– Adam PrzeździeckiJun 28 '17 at 12:10

$\begingroup$ I'll see what I can find more specifically (with precise references) in the next couple of days and then post an answer. This 13 January 2001 sci.math post might be of relevance, but I don't have time now to think about it (I'm at work now). $\endgroup$– Dave L RenfroJun 28 '17 at 14:01

$\begingroup$ @DaveLRenfro may be I wasn't precise enough  I am fully satisfied with your answer and want to reward it at least in reputation points. The reason for this silly question was that I am trying to find time to return to this topic after more than a year of inactivity and forgot about the lower packing dimension. $\endgroup$– Adam PrzeździeckiJun 28 '17 at 14:10

$\begingroup$ My interest in getting more specific is so that I when I want to look up something about this issue (or cite this answer at some later time), it will be more than just a pointer. Indeed, nearly every few days I find myself looking up something about a topic (that I know I've written about) by simply googling "Dave L. Renfro" along with one or more appropriate words/phrases. I'm pretty sure I've seen more than one paper that shows any choice of $5$ real numbers between $0$ and $n$ that doesn't violate an obvious inequality of any of these dimensions is possible for a single set. $\endgroup$– Dave L RenfroJun 28 '17 at 14:21
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 (19961997), 7073.
[2] Donald W. Spear, Sets with different dimensions in $[0,1]$, Real Analysis Exchange 24 #1 (19981999), 373389.
[3] Ying Xiong and Min Wu, Category and dimensions for cutout sets, Journal of Mathematical Analysis and Applications 358 #1 (1 October 2009), 125135.

$\begingroup$ The Math Forum link I used for that 13 Jan. 2001 sci.math post no longer works, but this googlegroups link currently works. However, in looking over that post, and the two earlier posts I made in the same sci.math thread, it's not clear to me now why I originally wrote "based on what I wrote" in the above answer. As far as I can tell now, that sci.math thread is not relevant to whether or not "examples showing that $h<m$ is possible" are in the two books I cited above (although I suspect such examples do exist there). $\endgroup$ Jun 2 '21 at 17:16