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Đề tài " Nonconventional ergodic averages and nilmanifolds "

We study the L2 -convergence of two types of ergodic averages. The first is the average of a product of functions evaluated at return times along arithmetic progressions, such as the expressions appearing in Furstenberg’s proof of Szemer´di’s theorem. | Annals of Mathematics Nonconventional ergodic averages and nilmanifolds By Bernard Host and Bryna Kra Annals of Mathematics 161 2005 397 488 Nonconventional ergodic averages and nilmanifolds By Bernard Host and BRyNA Kra Abstract We study the L2-convergence of two types of ergodic averages. The first is the average of a product of functions evaluated at return times along arithmetic progressions such as the expressions appearing in Furstenberg s proof of Szemeredi s theorem. The second average is taken along cubes whose sizes tend to to. For each average we show that it is sufficient to prove the convergence for special systems the characteristic factors. We build these factors in a general way independent of the type of the average. To each of these factors we associate a natural group of transformations and give them the structure of a nilmanifold. From the second convergence result we derive a combinatorial interpretation for the arithmetic structure inside a set of integers of positive upper density. 1. Introduction 1.1. The averages. A beautiful result in combinatorial number theory is Szemeredi s theorem which states that a set of integers with positive upper density contains arithmetic progressions of arbitrary length. Furstenberg F77 proved Szemeredi s theorem via an ergodic theorem Theorem Furstenberg . Let X X ạ T be a measure-preserving probability system and let A G X be a set of positive measure. Then for every integer k 1 1 A liminf A y A n T -nA n T-2nA n---n T knA 0 . N N n 1 It is natural to ask about the convergence of these averages and more generally about the convergence in L2 pf of the averages of products of bounded functions along an arithmetic progression of length k for an arbitrary integer k 1. We prove 398 BERNARD HOST AND BRYNA KRA Theorem 1.1. Let X X ụ T be an invertible measure-preserving probability system k 1 be an integer and let fj 1 j k be k bounded measurable functions on X. Then 1 lim 1 Nl fi Tnx f2 T2nx . fk Tknx N N n 0 .