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Andrew Best Dissertation.pdf (759.3 KB)

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Applications of Ergodic Theory to Number Theory and Additive Combinatorics

Best, Andrew
http://rave.ohiolink.edu/etdc/view?acc_num=osu1623938320486276

Abstract Details

2021, Doctor of Philosophy, Ohio State University, Mathematics.
This dissertation is devoted to the study of certain ergodic phenomena of number-theoretic and combinatorial flavor which occur in finite fields and in modular rings. Both the heuristics which guide this study and much of the methodology which implements it are sourced from the ergodic theory of integer actions. A classical result from the theory of finite fields says that any element in a sufficiently large finite field can be represented as a sum of two dth powers in the field. In Chapter 2, mathematical and historical aspects of this result are explored. The mathematical content of this chapter is two proofs of the result, one new and using the van der Corput differencing technique, and the other based on Fourier analysis and an application of a nontrivial estimate from the theory of finite fields. This chapter’s historical remarks sketch the evolution of mathematical knowledge in the vicinity of this problem and concern cyclotomy, Fermat’s last theorem, and diagonal equations. In Chapter 3, the Furstenberg–Sarkozy theorem and other fundamental results in ergodic theory are examined to develop a heuristic which suggests that theorems in ergodic theory that are true for totally ergodic systems should also hold, in some asymptotic sense, for sequences of rotations on modular rings. In Chapter 3 and then in Chapter 4, this heuristic is applied to formulate results pertaining, respectively, to single recurrence and to multiple recurrence. In both chapters, multiplicative obstructions from the presence of non-invertible elements in modular rings present obstacles of various difficulty. There are clear avenues to further research, especially leading from the results of Chapter 4.
Vitaly Bergelson (Advisor)
Daniel Thompson (Committee Member)
Alexander Leibman (Committee Member)
66 p.
Mathematics
total ergodicity; cyclic group; independent polynomials

Recommended Citations

Citations

  • Best, A. (2021). Applications of Ergodic Theory to Number Theory and Additive Combinatorics [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1623938320486276

    APA Style (7th edition)

  • Best, Andrew. Applications of Ergodic Theory to Number Theory and Additive Combinatorics. 2021. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1623938320486276.

    MLA Style (8th edition)

  • Best, Andrew. "Applications of Ergodic Theory to Number Theory and Additive Combinatorics." Doctoral dissertation, Ohio State University, 2021. http://rave.ohiolink.edu/etdc/view?acc_num=osu1623938320486276

    Chicago Manual of Style (17th edition)

osu1623938320486276
136
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This open access ETD is published by The Ohio State University and OhioLINK.