Convergence Questions in Ergodic Theory

本研究由两部分组成：沿子序列的几乎处处收敛和多重平均。在第一部分中，用Hardy场理论来描述良好的测量序列。基于过去二十年的工作，很明显，根据Hardy场对良好序列的完整刻画将是可能的-这在几乎所有问题中都是相当令人惊讶的发展。第二部分讨论多个遍历平均值。利用Hardy域可以完全刻画好序列的平均收敛和递归性质。对序列的随机建模也进行了研究。这项工作对几乎处处收敛的数学影响是基于傅立叶分析和数论。傅立叶分析的联系与Magyar、Stein、Wainger等人的工作有关。数论的联系是与整数的基有关的。这项工作对多重平均数的数学影响是在加法组合学中：与Szemerédi定理有关的问题。这些数学联系中的每一个都是PI先前建立的。与大多数理论数学工作一样，很难预测它对其他科学和人类其他活动的影响。但令人欣喜的是，这位提名人已经在物理学和计算机(硬件)安全方面找到了一些联系。他在热力学基础和硬件安全方面的合作是由于对后续遍历问题的研究而产生的。同样重要的是要注意到，该项目的活动与数学其他部分之间的联系已经导致了一些关于如何教授和观察一些基本数学理论的重要的教育见解。

This research project consists of two parts: almost everywhere convergence along subsequences and multiple averages. In the first part, the theory of Hardy fields is used to describe good sequences of measurements. Based on two decades of previous work, it has become clear that a complete characterization of good sequences in terms of Hardy fields will be possible - quite a surprising development in almost everywhere questions. The second part deals with multiple ergodic averages. It is suggested that complete characterizations of good sequences for mean convergence and recurrence are possible in terms of Hardy fields. Random modeling of sequences is also investigated.The mathematical impact of the work on almost everywhere convergence is on Fourier analysis and number theory. The Fourier analysis connection is related to the work of Magyar, Stein, Wainger and others. The number theory connection is with bases of the integers. The mathematical impact of the work on multiple averages is in additive combinatorics: problems related to Szemerédi's theorem. Each of these mathematical connections have been established previously by the PI. As with most theoretical mathematical work, it is difficult to predict the impact on other sciences and other activities of humanity. But quite pleasingly, the proposer has already found some connections in physics and computer (hardware) security. His collaborations in the foundation of thermodynamics and in hardware security have come about because of work on subsequence ergodic questions. It is also important to note that the connection between the project's activitiy and other parts mathematics has led to some important educational insights into how to teach and view some basic mathematical theories.