I'm reading a book named The Elements of Statistical Learning by Hastie, in section 2.5, Local Methods in High Dimensions, it says that the sampling density is proportional to $N^{\frac{1}{p}}$, where $p$ is the dimension of the input space and $N$ is the sample size.
I'm confused, what does it mean by sampling density? I do know, intuitively, as the dimension becomes larger, the sample becomes more sparse. But I don't understand exactly where does $N^{\frac{1}{p}}$ come from.
Recall that the sampling density or sampling rate is the number of recorded samples per unit distance.
Assume that a sample of size $N$ is taken from the unit cube $[0,1]^p$ in $\mathbb R^p$ (but the argument applies to every domain of finite volume). Roughly speaking, each point in the sample takes a volume $1/N$ of the domain, hence the distance between this point and its neighbors scales as $1/N^{1/p}$.