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Recurrence Properties of Measure-Preserving Actions of Abelian Groups and Applications

Ackelsberg, Ethan
http://orcid.org/0000-0002-0063-8229
http://rave.ohiolink.edu/etdc/view?acc_num=osu1657886359765668

Abstract Details

2022, Doctor of Philosophy, Ohio State University, Mathematics.
Recurrence and multiple recurrence in ergodic theory have deep connections with combinatorics, as observed by Furstenberg in his ergodic-theoretic proof of Szemerédi’s theorem on arithmetic progressions. Motivated by combinatorial problems in groups such as ℤd (the d-dimensional integer lattice), ⊕𝔽q (an infinite-dimensional vector space over a finite field 𝔽q), or (ℚ×,·) (the group of nonzero rational numbers under multiplication), we study recurrence properties of measure-preserving actions of countable discrete abelian groups. This dissertation is broken into two main parts. First, in Chapter 3, we juxtapose single recurrence with the dynamical properties of rigidity and weak mixing. We generalize several recent results about ℤ-actions to general abelian group actions and explore new phenomena that arise in the context of infinitely-generated groups and groups with torsion. Two significant applications presented in this chapter are: (1) constructing weakly mixing systems for which various number-theoretically meaningful sequences are rigidity sequences, and (2) establishing a distinction between recurrence and strong recurrence for abelian group actions. In the second, much longer, part, consisting of Chapters 4–8, we investigate the large intersections property for families of linear or polynomial functions. The results of this portion of the dissertation enhance, generalize, and unify many of the previously known results about large intersections for multiple recurrence. Using Furstenberg’s correspondence principle, we obtain a multitude of combinatorial applications dealing with such diverse situations as triangular patterns in ℤ2, geometric progressions in the integers, and polynomial configurations in rings of integers of number fields.
Vitaly Bergelson (Advisor)
Andrey Gogolev (Committee Member)
Nimish Shah (Committee Member)
288 p.
Mathematics
recurrence; rigidity sequence; ergodic average; characteristic factor

Recommended Citations

Citations

  • Ackelsberg, E. (2022). Recurrence Properties of Measure-Preserving Actions of Abelian Groups and Applications [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1657886359765668

    APA Style (7th edition)

  • Ackelsberg, Ethan. Recurrence Properties of Measure-Preserving Actions of Abelian Groups and Applications. 2022. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1657886359765668.

    MLA Style (8th edition)

  • Ackelsberg, Ethan. "Recurrence Properties of Measure-Preserving Actions of Abelian Groups and Applications." Doctoral dissertation, Ohio State University, 2022. http://rave.ohiolink.edu/etdc/view?acc_num=osu1657886359765668

    Chicago Manual of Style (17th edition)

osu1657886359765668
316
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This open access ETD is published by The Ohio State University and OhioLINK.