Convergence Questions in Ergodic Theory

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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.