On the Small Ball Inequality in All Dimensions

The point of interest is the dependence upon the logarithm of the volume of the rectangles. With n(d−1)/2 on the left above, the inequality is trivial, while it is conjectured that the inequality holdswith n(d−2)/2. This is known in the case of d = 2 (Talagrand, 1994), and a recent paper of two of the authors (Bilyk and Lacey, 2006) proves a partial result towards the conjecture in three dimensions. In this paper, we show that the argument of (Bilyk and Lacey, 2006) can be extended to arbitrary dimension. We also prove related results in the subjects of the Irregularity of Distribution, and Approximation Theory. The authors are unaware of any prior results on these questions in any dimension d ≥ 4.