Since the sample sequence is infinite, it is interesting to consider asymptotic bounds on dispersion. It is known that for and any metric, the best possible asymptotic dispersion is for points and dimensions . In this expression, is the variable in the limit and is treated as a constant. Therefore, any function of may appear as a constant (i.e., for any positive ). An important practical consideration is the size of in the asymptotic analysis. For example, for the van der Corput sequence from Section 5.2.1, the dispersion is bounded by , which means that . This does not seem good because for values of that are powers of two, the dispersion is . Using a multi-resolution Sukharev grid, the constant becomes because it takes a longer time before a full grid is obtained. Nongrid, low-dispersion infinite sequences exist that have ; these are not even uniformly distributed, which is rather surprising.
Steven M LaValle 2012-04-20