A Panorama of Discrepancy Theory

Discrepancy theory concerns the problem of replacing a continuous object with a discrete sampling. The roots of the theory are in H. Weyl’s fundamental work on uniformly distributed sequences. Just sequences have been sought for a long time and in 1935, J.G. van der Corput conjectured their non-existence. The conjecture was proved in 1945 by T. van Aardenne-Ehrenfest and then put in a quantitative form by her in 1949. In 1954, her quantitative result was improved by K.F. Roth, who observed that the study of the discrepancy of an infinite sequence in the unit torus T is equivalent to the study of the discrepancy of a finite set in T2, thereby introducing a geometric point of view in the study of irregularities of point distribution, or discrepancy theory. For many years, the main results were obtained by Roth and W.M. Schmidt. However, over the last 30 years, many mathematicians and computer scientists have developed the original theory and explored new directions and applications. Discrepancy theory is currently at a crossroads between number theory, combinatorics, Fourier analysis, algorithms and complexity, probability theory, and numerical analysis. Its current applications range from traditional science and engineering to modern computer science and financial mathematics.