I am working on Kenneth Falconer's book on fractal geometry. The book gives us 5 different equivalent definitions for the box dimension and these are the two needed for this question:
(i) Let $N_{\delta}(\boldsymbol{F})$ be the least number of sets with a diameter of at most $\delta$ that cover $\boldsymbol{F}$.
(v) Let $N_{\delta}(\boldsymbol{F})$ be the greatest number of disjoint balls of radius $\delta$ with centers in $\boldsymbol{F}$.
We want to use definition (i) to show that the upper box dimension of the von Koch curve is at most $\frac{\log(4)}{\log(3)}$ and definition (v) to show that the lower box dimension of the von Koch curve is at least $\frac{\log(4)}{\log(3)}$.
I think I might know how to do the first part: Let us call the von Koch curve with $\boldsymbol{K}$. Let $N_{\delta}(\boldsymbol{K})$ be the least number of sets with a diameter of at most $\delta$ that cover $\boldsymbol{K}$. Now if we let $3^{-k}\leq \delta \leq 3^{-k+1}$ then we see that $4^k$ sets give a cover for $\boldsymbol{K}$. So $4^{k} \geq N_{\delta}(\boldsymbol{K})$. So we get the inequality: $$\overline{dim_B}\boldsymbol{K}=\mathop{\overline{\lim}}_{\delta \to \ 0}\frac{log N_\delta(\boldsymbol{K})}{-log\delta}\leq \mathop{\overline{\lim}}_{k \to \ \infty}\frac{log4^{k}}{-log3^{-k}}=\frac{log4}{log3}$$
Is this correct? I don't know how to show the second part. Thanks for the help!