The ERWIN SCHRÖDINGER INTERNATIONAL INSTITUTE for MATHEMATICAL PHYSICS

Programme on "Combinatorics, Number Theory, and Dynamical Systems"

October 1 - November 30, 2011

organised by

Manfred Einsiedler Peter Grabner Christian Krattenthaler and Tamar Ziegler


Supported by the National Research Network "Analytic Combinatorics and Probabilistic Number Theory", a network financed by the Austrian Science Foundation FWF.
This programme will be followed by the programme "Algebraic versus Analytic Geometry", under the scientific direction of Herwig Hauser, János Kollár, Josef Schicho, and Duco van Straten.

Dynamical systems, which have been studied in their own right, have also been developed into a powerful tool to answer number theoretic and (especially infinite) combinatorial questions. On the other hand many questions concerning (discrete) dynamical systems are of a number theoretic or combinatorial nature. Prominent examples are Margulis' proof of the Oppenheim conjecture, or Furstenberg's proof of Szemerédi's theorem. Other examples are given by Bourgain's ergodic theorems on subsets of the integers, where methods from classical analytic number theory play an important rôle. The results of Green and Tao on the existence of arbitrarily long arithmetic progressions of prime numbers show the strength of the interplay between combinatorial, number theoretic, and dynamical methods and ideas. A further recent result in that context was the solution of the classical Gel'fond conjectures on the sum-of-digits of primes and squares by Mauduit and Rivat. The results of Adamczewski and Bugeaud, which give a lower bound for the complexity function of the digital expansion of algebraic irrationals, give a further prominent example for the combination of combinatorial and Diophantine techniques.

From this short description, it is clear that the three areas have a strong and fruitful interaction. The main aim of the proposed programme is to bring together researchers from these areas, in order to intensify this interaction.
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