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Topics in Ergodic Theory and Ramsey Theory

Farhangi, Sohail
http://orcid.org/0000-0001-8183-6157
http://rave.ohiolink.edu/etdc/view?acc_num=osu165203490336716

Abstract Details

2022, Doctor of Philosophy, Ohio State University, Mathematics.
This thesis is comprised of two main parts. The first part consists of topics in ergodic theory. In particular, we deal with variations of van der Corput's difference theorem, van der Corput sets, strengthenings of Birkhoff's ergodic theorem, and a generalization of the notion of uniform distribution. We show that van der Corput's difference theorem in Hilbert spaces and in uniform distribution theory is connected to the ergodic hierarchy of mixing properties. We show that our strengthening of Birkhoff's ergodic theorem leads to a variety of weighted ergodic theorems. We generalize the notion of uniform distribution to that of \textit{uniform symmetric distribution}, and obtain applications to measure preserving systems. We add to the known list of equivalent formulations of van der Corput sets, and answer some open questions from the literature. In the second part, which is based on joint work with Richard Magner, we give an almost complete classification of those $m,n \in \mathbb{N}$ and $a,b,c \in \mathbb{Z}\setminus\{0\}$ for which the equation $ax+by = cw^mz^n$ is partition regular over $\mathbb{Z}\setminus\{0\}$. This generalizes the result of Bergelson and Hindman about the partition regularity of the equation $x+y = wz$. One of the key ingrendients in the proof of our result is a partial generalization of the Grunwald-Wang Theorem. We also prove results of independent interest about ultrafilters $q$ over an infinite integral domain $R$ for which each $A \in q$ has substantial additive and multiplicative structure.
Vitaly Bergelson (Advisor)
Ilya Gruzberg (Committee Member)
Alexander Leibman (Committee Member)
Andrey Gogolyev (Committee Member)
212 p.
Mathematics
van der Corput, pointwise ergodic theorem, uniform distribution, ramsey theory, ultrafilters, recurrence

Recommended Citations

Citations

  • Farhangi, S. (2022). Topics in Ergodic Theory and Ramsey Theory [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu165203490336716

    APA Style (7th edition)

  • Farhangi, Sohail. Topics in Ergodic Theory and Ramsey Theory. 2022. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu165203490336716.

    MLA Style (8th edition)

  • Farhangi, Sohail. "Topics in Ergodic Theory and Ramsey Theory." Doctoral dissertation, Ohio State University, 2022. http://rave.ohiolink.edu/etdc/view?acc_num=osu165203490336716

    Chicago Manual of Style (17th edition)

osu165203490336716
386
© 2022, some rights reserved.Creative Commons License Topics in Ergodic Theory and Ramsey Theory by Sohail Farhangi is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Based on a work at etd.ohiolink.edu.
This open access ETD is published by The Ohio State University and OhioLINK.