Low-discrepancy sequence

Summary

A low-discrepancy sequence is a sequence of points in the unit cube $[0, 1)^s$ whose initial segments fill the cube as evenly as possible. Evenness is measured by the star discrepancy. It compares the share of the first $N$ points in each box anchored at the origin with the volume of the box. For a low-discrepancy sequence the largest difference is of order $(\log N)^s / N$. For large $N$ this is much smaller than the typical value of order $N^{-1/2}$ for random points. Such sequences are the point sets of quasi-Monte Carlo integration. Every low-discrepancy sequence is uniformly distributed in the cube; for $s = 1$ this is equidistribution modulo one.