Applications of Ergodic Theory to Combinatorics and Number Theory

本课题主要研究遍历理论中的多重递归问题，重点研究其与组合学和数论的相互丰富的联系。所考虑的问题可以看作是动力学中经典递归结果的深远扩展。同时，这些问题也使得遍历理论在组合学、数论和代数中得到了广泛的应用，这是迄今为止传统方法所无法达到的。近年来提出的一些多项式结果推动了多次递归理论的进一步发展。这些发展提供了对多项式多次递归现象的更好理解，并带来了新的研究前景。其中一些远景导致了与素数和有限组合理论的有趣的新联系和应用。本建议考虑的新问题反映了新方法和新技术的进入。这些方法包括利用零流形上的动力系统的几何方法和涉及斯通-切赫紧化的拓扑代数方法。我们所熟悉的关于交换群的结果不仅可以很自然地推广到幂零的情形，而且幂零动力学还允许我们得到关于保测度变换的阿贝尔群的收敛性和递归性的新信息。该提案中提出的一些猜想为幂零动力学与遍历理论和组合学的重要问题之间的联系提供了新的线索。另一组猜想是利用拓扑代数技术来改进递归结果的。然而，另一组猜想是基于新的结果和思想，它们将遍历论、有限特征函数域中的数论和组合学联系在一起。遍历拉姆齐理论领域的问题，技术和应用的多样性是一个很好的媒介，吸引本科生到数学和研究生到一个活跃的研究领域。在这个建议中提出的问题和猜想连接了数学的不同领域（遍历论、组合学、代数、数论），并对每个领域都有所贡献。例如，近年来，起源于多次递归理论的方法、结果和思想（其中一些是由提出者提出的）给素数理论带来了惊人的进步。近年来出现的另一个有趣的研究方向是将有限域的多次递归遍历理论与组合学联系起来。这些发展与理论计算机科学有关。提出的研究旨在更好地理解在多项式（和更一般）函数值对应的时刻采样的动力系统行为的规律性。虽然该提案侧重于这种现象在组合学和数论中的应用，但物理学家也可能对此感兴趣。

The project is focused on the problems of multiple recurrence in ergodic theory with emphasis on the mutually enriching connections with combinatorics and number theory. The problems considered may be viewed as far reaching extensions of classical recurrence results in dynamics. At the same time, these problems lead to strong applications of ergodic theory to combinatorics, number theory and algebra which are inaccessible, so far, by conventional methods. Some of the polynomial results obtained by the proposers in recent years served as an impetus for further developments in the theory of multiple recurrence. These developments provide better understanding of the phenomenon of multiple recurrence along polynomials and bring new vistas of research to light. Some of these vistas lead to interesting new connections with and applications to the theory of prime numbers and finitary combinatorics. The new problems considered in this proposal reflect entrance of new methods and techniques into the picture. These include the geometric method utilizing dynamical systems on nilmanifolds and methods involving the topological algebra in Stone-Cech compactifications. Not only are most of the familiar results dealing with commutative groups naturally extendible to the nilpotent setup, but also it turns out that nilpotent dynamics allows one to get new information about convergence and recurrence properties of abelian groups of measure preserving transformations. Some of the conjectures formulated in the proposal shed new light on the connections of nilpotent dynamics with important problems of ergodic theory and combinatorics. Another group of conjectures deals with refining of the recurrence results with the help of the topological algebra techniques. Yet another group of conjectures is based on new results and ideas which link together ergodic theory, number theory in function fields of finite characteristics, and combinatorics.The area of Ergodic Ramsey Theory with its diversity of problems, techniques and applications is an excellent medium for attracting undergraduates to mathematics and graduate students to an area of active research. The problems and conjectures that are posed in this proposal connect diverse areas of mathematics (ergodic theory, combinatorics, algebra, number theory) and contribute to each. For example, in recent years the methods, results and ideas originating in the theory of multiple recurrence (some of which are due to proposers) have brought spectacular advancements in the theory of prime numbers. Another interesting direction of research that have emerged in recent years links together ergodic theory of multiple recurrence with combinatorics in finite fields. These developments have connections with the theoretical computer science. The proposed study aims at better understanding of the regularity of the behavior of dynamical systems sampled at moments of time corresponding to values of polynomial (and more general) functions. While the proposal focuses on applications of this phenomenon in combinatorics and number theory, it may be of interest to a physicist as well.