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The dichotomy between structure and randomness and applications to combinatorial number theory

Richter, Florian K
http://rave.ohiolink.edu/etdc/view?acc_num=osu1532042074011706

Abstract Details

2018, Doctor of Philosophy, Ohio State University, Mathematics.
The study of the long-term behavior of dynamical systems has far-reaching applications to other areas of mathematics. The employment of analytic tools coming from measurable, topological, and symbolic dynamics offers novel possibilities for analyzing seemingly static number-theoretic and combinatorial situations and has proven to be a powerful method in solving numerous open problems in Ramsey theory and combinatorial number theory that previously appeared to be intractable. In this thesis we develop new techniques that are inspired by dynamical heuristics and lead to a variety of applications in discrete mathematics. One theme featured prominently in this work is the idea of dichotomy between structure and randomness. This dichotomy manifests itself via decomposition theorems that deal with splittings of arithmetic functions into two components, one of which is structured and the other is pseudo-random. From these decomposition theorems we then derive results in ergodic theory and density Ramsey theory. Among other things, we obtain generalizations and refinements of Szemeredi’s theorem and Sarkozy’s theorem, and present a solution to a long-standing open sumset conjecture of Erdos.
Bergelson Vitaly (Advisor)
Leibman Alexander (Committee Member)
Penneys David (Committee Member)
129 p.
Mathematics

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Citations

  • Richter, F. K. (2018). The dichotomy between structure and randomness and applications to combinatorial number theory [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1532042074011706

    APA Style (7th edition)

  • Richter, Florian. The dichotomy between structure and randomness and applications to combinatorial number theory. 2018. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1532042074011706.

    MLA Style (8th edition)

  • Richter, Florian. "The dichotomy between structure and randomness and applications to combinatorial number theory." Doctoral dissertation, Ohio State University, 2018. http://rave.ohiolink.edu/etdc/view?acc_num=osu1532042074011706

    Chicago Manual of Style (17th edition)

osu1532042074011706
588
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This open access ETD is published by The Ohio State University and OhioLINK.