Dispersion bounds

Since the sample sequence is infinite, it is interesting to consider asymptotic bounds on dispersion. It is known that for $X = [0,1]^n$ and any $L_p$ metric, the best possible asymptotic dispersion is $O(k^{-1/n})$ for $k$ points and $n$ dimensions [738]. In this expression, $k$ is the variable in the limit and $n$ is treated as a constant. Therefore, any function of $n$ may appear as a constant (i.e., $O(f(n)k^{-1/n}) = O(k^{-1/n})$ for any positive $f(n)$). An important practical consideration is the size of $f(n)$ in the asymptotic analysis. For example, for the van der Corput sequence from Section 5.2.1, the dispersion is bounded by $1/k$, which means that $f(n) = 1$. This does not seem good because for values of $k$ that are powers of two, the dispersion is $1/2k$. Using a multi-resolution Sukharev grid, the constant becomes $3/2$ because it takes a longer time before a full grid is obtained. Nongrid, low-dispersion infinite sequences exist that have $f(n) = 1/\ln 4$ [738]; these are not even uniformly distributed, which is rather surprising.