Mathematics

This seminar aims to introduce and discuss tools in the theory of irregularities of distributions. The definition of uniform distribution for a sequence in the one-dimensional torus is common knowledge, but for this qualitative attribute, we can associate a quantitative counterpart called discrepancy. At this point, we will present a fundamental theorem in this field: K.F. Roth's Theorem (recognized as his favourite theorem, despite the one that got him the Fields medal in 1958). Then, we will introduce the two most important sequences in this field, namely the van Der Corput sequence and the Kronecker sequence. Finally, we will conclude by showing the theorem of K. Petersen on Kronecker sequences, which answers a conjecture of P. Erdös and P. Szüsz through ergodic theory techniques.