To set the stage, let me recall the definition of the box-counting dimension of a set $F \subset \mathbb R^n$.
The lower and upper box-counting dimensions of a subset $F$ of $ℝ^n$ are given by
$$\underline{\dim}_B F = \underline{\lim}_{\delta\to 0} \frac{\log N_\delta(F)}{-\log \delta}$$
$$\overline{\dim}_B F = \overline{\lim}_{\delta\to 0} \frac{\log N_\delta(F)}{-\log \delta}$$
and the box-counting dimension of $F$ by
$$\dim_B F = \lim_{\delta\to 0} \frac{\log N_\delta(F)}{-\log \delta}$$
(if this limit exists), where $N_\delta(F)$ is the smallest number of sets of diameter at most $𝛿$ that cover $F$.
Now, the author says that to find $\dim_B F$, it is enough to consider limits as $𝛿\to 0$ through $\{𝛿_k\}$ such that $𝛿_{k+1} \ge c𝛿_k$ for some $0 < c < 1$, in particular $\delta_k = c^k$. I have trouble seeing why this is true. Following is the reasoning provided by the author:
Note that if $\delta_{k+1} \le \delta < \delta_k$, then with $N_𝛿(F)$ the least number of sets in a $𝛿$-cover of F,
$$\frac{\log N_\delta(F)}{-\log\delta} \le \frac{\log N_{\delta_{k+1}}(F)}{-\log\delta_k} = \frac{\log N_{\delta_{k+1}}(F)}{-\log\delta_{k+1} + \log{(\delta_{k+1}/\delta_k)}} \le \frac{\log N_{\delta_{k+1}}(F)}{-\log\delta_{k+1} + \log c}$$
and so
$$\overline{\lim_{\delta\to 0}} \frac{\log N_\delta(F)}{-\log\delta} \le \overline{\lim_{k\to\infty}} \frac{\log N_{\delta_k}(F)}{-\log\delta_k}$$
The opposite inequality is trivial; the case of lower limits may be dealt with in the
same way.
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The author says $0 < c < 1$, so that $\log c < 0$. I do not understand where the following implication comes from:
$$\frac{\log N_\delta(F)}{-\log\delta} \le \frac{\log N_{\delta_{k+1}}(F)}{-\log\delta_{k+1} + \log c} \implies \overline{\lim_{\delta\to 0}} \frac{\log N_\delta(F)}{-\log\delta} \le \overline{\lim_{k\to\infty}} \frac{\log N_{\delta_k}(F)}{-\log\delta_k} $$ -
For the opposite direction (assuming the implication in (1) is correct), I would do
$$\frac{\log N_\delta(F)}{-\log\delta} \ge \frac{\log N_{\delta_{k}}(F)}{-\log\delta_{k+1}} = \frac{\log N_{\delta_{k}}(F)}{-\log\delta_{k} + \log(\delta_k/\delta_{k+1})} \ge \frac{\log N_{\delta_{k}}(F)}{-\log\delta_{k} – \log c}$$
Taking limits (this needs more justification),
$$\overline\lim_{\delta\to 0} \frac{\log N_\delta(F)}{-\log\delta} \ge \overline\lim_{k\to\infty} \frac{\log N_{\delta_k}(F)}{-\log\delta_k}$$
which is the required inequality. Combining this with the previous inequality, stated in the question, we get $$\overline\lim_{\delta\to 0} \frac{\log N_\delta(F)}{-\log\delta} = \overline\lim_{k\to\infty} \frac{\log N_{\delta_k}(F)}{-\log\delta_k}$$
A similar procedure could be adopted for the case of $\underline\lim_{\delta\to 0}$ and $\underline\lim_{k\to \infty}$ (just take lower limits in place of upper ones, everything else unchanged). Is this right?
References.
- Fractal Geometry by Kenneth Falconer.
Best Answer
Below is a proof I have for the lower box dimension, from some handwritten notes I wrote in January 1991. I don't have time now to check the details, other than a quick proof-reading of the expressions, so let me know if something seems amiss and I'll try to address your concerns at some later time.
Let $\delta > 0$ be given. Then there exists $k$ such that $\delta_{k+1} < \delta < \delta_{k}.$ Therefore,
$$\frac{\log N_\delta(F)}{-\log\delta} \;\; \leq \;\; \frac{\log N_{\delta_{k+1}}(F)}{-\log\delta} \;\; \leq \;\; \frac{\log N_{\delta_{k+1}}(F)}{-\log\delta_k} \;\; = \;\; \frac{\log N_{\delta_{k+1}}(F)}{-\log\left(\frac{\delta_k}{\delta_{k+1}} \cdot \delta_{k+1}\right)} $$
$$ = \;\; \frac{\log N_{\delta_{k+1}}(F)}{-\log\left(\frac{\delta_k}{\delta_{k+1}}\right) - \log \delta_{k+1}} \;\; \leq \;\; \frac{\log N_{\delta_{k+1}}(F)}{\log c - \log \delta_{k+1}}, $$
where for the last inequality note that
$$\frac{\delta_{k+1}}{\delta_k} \geq c \;\; \implies \;\; \log \frac{\delta_k}{\delta_{k+1}} \leq \log \frac{1}{c} = -\log c \;\; \implies \;\; \log c \leq -\log\left(\frac{\delta_k}{\delta_{k+1}}\right),$$
and so using $\log c$ in "the last inequality" results in a smaller denominator (hence, a larger value).
Therefore,
$$ \liminf_{\delta \rightarrow 0^+}\, \frac{\log N_{\delta}(F)}{-\log \delta} \;\; \leq \;\; \liminf_{k \rightarrow \infty}\, \frac{\log N_{\delta_{k+1}}(F)}{\log c - \log \delta_{k+1}} $$
$$= \;\; \liminf_{k \rightarrow \infty}\, \left[ \frac{\;\;\frac{\log N_{\delta_{k+1}}(F)}{-\log \delta_{k+1}}\;\;}{\frac{\log c}{-\log \delta_{k+1}} + 1} \right] \;\; = \;\; \liminf_{k \rightarrow \infty}\, \frac{\log N_{\delta_{k+1}}(F)}{-\log \delta_{k+1}} \;\; = \;\; \liminf_{k \rightarrow \infty}\, \frac{\log N_{\delta_k}(F)}{-\log \delta_k}, $$
where in the second to last equality above we are using the fact that $\frac{\log c}{-\log \delta_{k+1}} \rightarrow 0$ as $k \rightarrow \infty.$
Interchanging $k$ with $k+1$ in the above is easily seen to reverse each inequality, and so the other inequality also holds.