I am trying to find the lower- and upper-box dimensions (and show that they are the same) of the set $A=\{0,1,\frac{1}{4},\frac{1}{9},\ldots\}=\{\frac{1}{n^{2}}:n\in\mathbb{Z}_{\geqslant0}\}\cup\{0\}$. My thinking: There are $k$ intervals of length $\frac{1}{k^{2}}$ at stage $k$ of the construction. So $$\dim_{B}(A)=\lim_{\varepsilon\to0}\frac{\log{N_{\varepsilon}(A)}}{-\log{\delta}}=\lim_{k\to\infty}\frac{\log{k}}{-\log{k^{2}}}=\lim_{k\to\infty}\frac{\log{k}}{2\log{k}}=0.5.$$ But this doesn’t feel right. I haven’t found the upper- and lower- limits, I have just kind […]

It occurs to me that people most likely already know how to explicitly integrate over fractals, but my method (edit: seems to have been highlighted out in a paper, see comments) seems to vastly simplify the process (So even a comparative layman like me can do it). Firstly, we’ll be integrating with respect to measure, […]

That’s just a fun question. Please, be creative. Suppose having a Koch snowflake. The area inside this curve is having the total mass $M$ and the length of the first iteration is $L$ (a simple equilateral triangle of side $L$). Assume the density being uniform. If the curve is on $xy$ plane, the rotation is […]

What is $$\int_{K} e^{a \cdot x+ b \cdot y} \mu(x,y)$$ where $K$ is the Koch curve and $\mu(x,y)$ is a uniform measure look here. Attempt: I can evaluate the integral numerically and I have derived a method to integrate $e^x$ over some cantor sets, look here. When I tried using that method to integrate the […]

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