Discrepancy theory and harmonic analysis

In the present survey we discuss various applications of methods and ideas of harmonic analysis in problems of geometric discrepancy theory and irregularities of distribution. A great number of analytic tools (exponential sums, Fourier series, Fourier transform, orthogonal expansions and wavelets, Riesz products, Littlewood–Paley theory, Carleson’s theorem) have found applications in this area. Some of the methods have been used since the birth of the subject, while the more modern ideas are still paving their way into the field. We illustrate their applications by considering several standard topics in uniform distribution theory: Weyl’s criterion, metric estimates for the discrepancy of sequences, discrepancy with respect to balls and rotated cubes, lower bounds for the discrepancy function. This exposition of Fourier-analytic techniques in discrepancy theory is intended for a broad mathematical readership.