Analytic Combinatorics and Probabilistic Number Theory

## Principal InvestigatorReinhard Winkler## Co-InvestigatorGerhard Dorfer## Funded Researchers
Mathias
Beiglböck (on leave) ## DescriptionAlthough number theory and combinatorics have originally dealt with questions of discrete nature, analytic methods have turned out to be extremely powerful for their solution. Three classical examples: At the end of the 19th century the first proofs of the prime number theorem were based on complex analysis. In 1916 Hermann Weyl's approach to uniform distribution of sequences via harmoic analysis opened the way to ergodic theory with very strong impact on dynamical systems. And in the 70s of the 20th century Furstenberg and others found fascinating applications of ergodic theory and topology to combinatorial number theory in the spirit of the theorems of van der Waerden and Szemerédi. In this project we focus on the measure theoretic and topological point of view of some number theoretical problems. We will use harmonic analysis to study certain coding sequences from symbolic dynamics. This is closely related to the connections between sequences of integers (or characters) and subgroups of the torus (or other compact groups) which we want to investigate. In this context, as well as for questions on the distribution of sequences and subsequences or on Diophantine approximation, we expect that Baire categories will play an important role. We hope to obrtain further results in combinatirial number theory by use of the Stone-Čech and maybe other comactifications. Finally, the investigation of the interplay between digital representations and topology is intended to give new insights into the structure of fractals arising from dynamical systems of number theoretic origin. |