The box counting method of measuring the fractal dimension of an object is $$D = \lim_{\epsilon \rightarrow 0}{ {\log N( \epsilon)} \over {\log { {1}\over{ \epsilon }}}},$$ where the classic example is calculating fractal dimension of the coast of England.
But let us imagine a scenario where the coast goes off to infinity in the north and south dimension. Let us fix $\epsilon$, and let $N$ be the independent variable, and let $R_N$ be radius of the smallest circle that contains $N$ boxes. Can we not write
$$D = \lim_{N \rightarrow \infty}{{\log { R_N}}\over {\log N} }.$$
It seems to me this is a perfectly valid method, but I cannot find it anywhere.