Local statistics of arithmetic sequences

A fundamental observation in mathematics asserts that many deterministic (that is, non-random) sequences of numbers exhibit properties which are typical for purely random sequences. For example, the distribution of prime numbers has many such random properties, although the prime numbers themselves are of course defined in a purely deterministic way. In this research project we will study such randomness and non-randomness phenomena, on a local level (unfortunately, the term local cannot be explained in more detail here). The appropriate random model is the Poisson process, a random process which is for example used to model the arrival of callers in a telephone hotline. Astonishingly, the local distribution of many deterministic sequences of number-theoretic origin coincides exactly with the distribution of this Poisson process, while in a small number of exceptional cases there is no such accordance. Remarkably, a similar phenomenon can be observed in quantum physical models in theoretical physics. In our research project we will investigate which systems follow the distribution of the Poisson process, which do not, and why this is the case. The topic covers many distinct mathematical areas: number theory, probability theory, analysis, ergodic theory, and has close connections with theoretical physics.

This project was concerned with fundamental mathematical research on topics including local statistical properties of sequences of arithmetic origin. Together with the co-applicant Daniel El-Baz (PostDoc researcher) we worked intensively on the problems suggested in the proposal, and obtained some very interesting results. Some of the results of this project were published in mathematical journals of the highest level such as Journal of the European Mathematical Society or Composito Mathematica. Furthermore, a PhD student was educated in this project.