I am putting the finishing touches on my master's essay for graduation this semester and I want to end my paper with a proof of why Hausdorff dimension and Minkowski (box) dimension are different.
I haven't found the proof in Fractal Geometry: Mathematical Foundations and Applications by Kenneth Falconer, nor have I found anything online either. I found many places where the assertion is made that they are not equivalent, but nowhere do these sources give more than an expository explanation. Where can I find this proof ? Thank you.
The simplest example is a dense set like the rationals in the unit interval, that has Hausdorff dimension 0 and Minkowski dimension 1. A closed set where the dimensions differ is $\{0\} \cup \{1/n\}_{n \ge 1}$; here the Hausdorff dimension is 0 and the Minkowski dimension is $1/2$. Chapter 1 in the book [1] discusses the comparison. Very interesting sets where the two notions of dimension differ are the Self-affine carpets of McMullen and Bedford, studied in Chapter 4 of [1].
[1] https://www.cambridge.org/core/books/fractals-in-probability-and-analysis/D8CBD4181FDC20C387E22939DA2F6168#fndtn-information available at https://www.math.stonybrook.edu/~bishop/classes/math324.F15/book1Dec15.pdf