Geometric features of Fourier transform and applications to discrepancy theory

This thesis deals with the interplay between harmonic analysis and geometry. We present novel relationships between the Fourier transform and the geometric properties of sets, and more generally of functions of bounded variation. These are then applied to the theory of irregularities of distribution, often referred to as discrepancy theory, which quantifies how point distributions in a space deviate from uniformity. We study quadratic averages of the discrepancy over families of sets generated by the similarity group, for three different choices of the set of rotations averaged over.