Polynomial ergodic theorems and Ramsey theory

摘要：本课题主要研究动力系统中的多重递归收敛问题，重点研究阿贝尔群和幂零群的多项式作用。所考虑的问题导致遍历理论在组合学、数论和代数上的各种应用，这些应用迄今为止是传统方法无法达到的。近年来，由提出者证明的多项式Szemeredi定理和hales - jewett定理被提出者及其同事在不同的方向上进行了扩展，每个定理都提供了对多项式多次递归现象的更好理解，并提供了新的研究前景。本文涉及的研究方向包括（但不限于）幂零群作用的多重递归的深入研究和ip系统框架下多项式多重递归的研究。在这个建议中考虑的另一个有趣的领域与遍历平均的收敛有关，它自然地出现在多次递归理论及其应用中。作者最近得到的结果表明，遍历平均的行为取决于作用群是多项式增长还是指数增长。在提案的最后一节中提出的相关猜想为这些问题提供了新的线索。我们在这个项目中关注的多重递归理论使用了几个不同数学领域的思想，旨在提高我们对动力系统的内在特性的认识，这些特性与它们的长期行为有关。从我们的研究中产生的一个非常重要的事实的例子是动力系统沿多项式时间测量的行为的规律性。这个事实，反过来，在看似遥远的数论和组合学领域有很强的应用。

ABSTRACT:The project concentrates on multiple recurrence and convergence indynamical systems with a focus on polynomial actions of abelian andnilpotent groups. The problems considered lead to diverse applications ofergodic theory to combinatorics, number theory and algebra which areinaccessible so far by conventional methods. The polynomial Szemeredi andHales-Jewett theorems proved by the proposers in the recent years havesince been extended by the proposers and their colleagues in differentdirections, each providing a better understanding of the phenomenon ofmultiple recurrence along polynomials and offering new vistas of research.The directions of study touched upon in this proposal include(but are not limited to) the deeper study of multiple recurrence fornilpotent group actions and the investigation of polynomial multiplerecurrence in the framework of IP-systems. Another interesting areaconsidered in this proposal has to do with convergence of ergodic averagesnaturally appearing in the theory of multiple recurrence and itsapplications. The results recently obtained by the proposers indicate adichotomy between the behavior of ergodic averages depending on whetherthe acting groups have polynomial or exponential growth. Therelated conjectures formulated in the last section of the proposal areshedding new light on these questions. The theory of multiple recurrence that we focus on in this project usesthe ideas from several diverse areas of mathematics and aims to advanceour knowledge about the intrinsic properties of dynamical systems whichare related to their long range behavior. An example of a highlynontrivial fact stemming from our investigations is the regularity of thebehavior of dynamical systems along polynomial time measurements. Thisfact, in its turn, has strong applications to seemingly distant areas ofnumber theory and combinatorics.