Irregularities of Distribution and Average Decay of Fourier Transforms

In Geometric Discrepancy we usually test a distribution of N points against a suitable family of sets. If this family consists of dilated, translated and rotated copies of a given d -dimensional convex body D contained in the unit box, then a result proved by W. Schmidt, J. Beck and H. Montgomery shows that the corresponding L2 discrepancy cannot be smaller than c_d*N^{(d-1)/(2d)} . Moreover, this estimate is sharp, thanks to results of D. Kendall, J. Beck and W. Chen. Both lower and upper bounds are consequences of estimates of the decay for large values of the frequency variable, of the L2 norm with respect to rotations of the characteristic function of the convex body D. In this chapter we provide the Fourier analytic background and we carefully investigate the relation between the L2 discrepancy and the estimates of the above mentioned L2 norm.