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I am working through a first course in Fractal Geometry, and have encountered a problem that has asked me to calculate the box-counting dimension of $F=\{ \frac{1}{5^n} : n \in \mathbb{N}\}$

However, I am stuck straight away. Thus far, I've only encountered problems that have asked me to calculate the box-counting dimension of things like the Van Koch curve or variants of it. So I'm struggling on how to approach this since we only have a set of points, as opposed to a continuous line.

If anyone could perhaps give me a nudge in how I should tackle this, it would be much appreciated. :)

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  • $\begingroup$ Well you could just count how many intervals of $\{ [\frac k n , \frac{k+1} n], 0 \leq k \leq n -1 \}$ are needed to recover $F$, and take the limit for $n \rightarrow \infty$. It might be easier to consider only a subset of these : maybe $\{ [\frac{k}{5^n} , \frac{k+1}{5^n}], 0 \leq k \leq 5^n -1 \}$ or something like that. $\endgroup$
    – sapristi
    Nov 12, 2015 at 21:06

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For a sequence decreasing exponentially to zero like this, you might expect the dimension to be zero. To prove it, you need an upper bound on $N_{\varepsilon}(F)$, the number of intervals of length $\varepsilon$ required to cover $F$, for then you can compute $$\dim(F) = \lim_{\varepsilon\searrow0}\frac{\log(N_{\varepsilon}(F))}{\log(1/\varepsilon)}.$$

To this end, let $\varepsilon > 0$ and let $n$ be the unique positive integer such that $$\frac{1}{5^{n+1}} \leq \varepsilon < \frac{1}{5^n}.$$ Since $\varepsilon \geq 1/5^{n+1}$, the interval $[0,\varepsilon]$ covers all but $n$ of the points in $F$. Thus $N_{\varepsilon}(F) \leq n+1$. Also, since $\varepsilon < 1/5^n$, $$\frac{1}{\log(1/\varepsilon)} < \frac{1}{\log(5^n)}.$$ Putting these inequalities together, we get $$\frac{\log(N_{\varepsilon}(F))}{\log(1/\varepsilon)} \leq \frac{\log(n+1)}{n\log(5)} \to 0$$ as $n\to \infty$.

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