Low-Discrepancy Sets for Axis-Parallel Boxes

Abstract

What should a planar set with small discrepancy for axis-parallel rectangles look like? Maybe the first thing coming to mind would be the regular $\sqrt n \times \sqrt n$ grid, placed in the unit square in an appropriate scale, as in Fig. 2.1(a). It is easy to see that this gives discrepancy of the order $\sqrt n$. Another attempt might be n independent random points in the unit square as in Fig. 2.1(b), but these typically have discrepancy about $\sqrt n$ as well. (In fact, with high probability, the discrepancy is of the order $\sqrt {n\;\log \;\log \;n}$; a result well-known to probabilists under the name law of the iterated logarithm comes into play.) It turns out that a far better discrepancy can be achieved, of the order log n. This chapter is devoted to various constructions of such sets and to their higher-dimensional generalizations. In dimension d, for d arbitrary but fixed, the best known sets have discrepancy for axis-parallel boxes of the order log^d−1 n.