Ergodic Theory, Differential Dynamics and Combinatorial Number Theory

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This proposal contains a number of different problems whose common thread is a core approach combining dynamics, probability theory, and a basic set of techniques that were mainly developed in the classical ergodic theoretic setting. These techniques have been very successfully adapted to problems from hard analysis and combinatorics, in areas such as: KAM theory; Ramsey theory (in particular density versions of results previously known for partitions, e.g., the density version of the Hales-Jewett theorem); Fractal geometry and Haussdorff dimension of sets in Euclidean space; the theory of amenable groups and their actions. These are the main areas that we propose to develop further.Physical systems satisfying the same set of laws exhibit a wide variety of behaviors going from the extreme of completely chaotic systems which appear to be completely random to systems that exhibit a surprising level of stability (for instance, a particle moving in a cyclotron). Mathematical dynamics and ergodic theory explain these different types of behaviors. The same methods turn out to be extremely useful in other areas, in particular in combinatorics, information theory, data compression, etc. Our work is developing new methods that give better explanations of these phenomena and extend the applicability of the field to a broader class of problems.