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Quasirandom Groups, Minimally Almost Periodic Groups and Ergodic Ramsey Theory

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Interactions between Dynamics of Group Actions and Number Theory

According to the definition introduced by T. Gowers in 2008, a finite group G is called D-quasirandom for some parameter D, if all non-trivial unitary representations of G have dimension greater or equal to D. For example, the group SL(2, F_p) is (p-1)/2 quasirandom for any prime p. Informally, a finite group is quasirandom if it is D-quasirandom for a large value of D. Answering a question posed by L. Babai and V. Sos, Gowers have shown that, in contrast with the more familiar “abelian” situation, qusirandom groups can not have large product-free subsets. The goal of this lecture is to discuss the connection between the combinatorial phenomena observed in quasirandom groups and the ergodic properties of the minimally almost periodic groups (these were introduced in 1934 by J. von Neumann as groups which do not admit non-constant almost periodic functions). This connection will allow us to give a simple explanation the dynamical underpinnings of some of the Gowers’ results as well as of the more recent results obtained in joint work with T. Tao and in the work of T. Austin.

This talk is part of the Isaac Newton Institute Seminar Series series.

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