Convergence Questions in Ergodic Theory

本研究计画包含两部分：几乎处处沿着连续收敛与多重平均。第一部分用哈代场理论描述好的测量序列。基于过去20年的工作，我们已经清楚地看到，用哈代场来完全刻画好的序列是可能的--这在几乎所有的问题中都是一个令人惊讶的发展。第二部分处理多个遍历平均值。 本文提出了用哈代域完全刻画好的序列的平均收敛性和常返性。随机建模的序列也调查。数学影响的工作几乎处处收敛是傅立叶分析和数论。傅立叶分析的连接是有关的工作马扎尔，斯坦，Wainger和其他人。数论与整数的基有关。数学上的影响工作的多重平均是在添加剂组合学：问题有关塞梅雷迪的定理。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。 但令人高兴的是，提议者已经发现了物理学和计算机（硬件）安全的一些联系。他的合作基础上的热力学和硬件安全已经因为工作的后续遍历问题。同样重要的是要注意到，该项目的活动和其他部分之间的联系数学已经导致了一些重要的教育见解如何教和查看一些基本的数学理论。

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.