Asymptotic bounds

There are many different asymptotic bounds for discrepancy, depending on the particular range space and measure space [682]. The most widely referenced bounds are based on the standard range space of axis-aligned rectangular boxes in $[0,1]^n$. There are two different bounds, depending on whether the number of points, $k$, is given. The best possible asymptotic discrepancy for a single sequence is $O(k^{-1}\log^n k)$. This implies that $k$ is not specified. If, however, for every $k$ a new set of points can be chosen, then the best possible discrepancy is $O(k^{-1}\log^{n-1} k)$. This bound is lower because it considers the best that can be achieved by a sequence of points sets, in which every point set may be completely different. In a single sequence, the next set must be extended from the current set by adding a single sample.