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Algebraic, analytic, and geometric notions of largeness for subsets of Zd and their applications

Glasscock, Daniel G
http://rave.ohiolink.edu/etdc/view?acc_num=osu1499952794190004

Abstract Details

2017, Doctor of Philosophy, Ohio State University, Mathematics.
In this dissertation, we investigate the relationships between and applications of several different notions of largeness for subsets of Zd, the d-dimensional integer lattice, and, more generally, groups and semigroups. The various meanings of the word "large" appearing here are inspired by ideas in algebra, analysis, and geometry and are significant for their historical influence and usefulness in applications to combinatorics and number theory. This work is comprised of three main parts. In the first, we study the relationships between several naturally occurring notions of additive and multiplicative largeness in rings and semirings. We harness these relationships and the ideas behind them to give a number of novel applications at the intersection of Ramsey theory and number theory. In the second part, we introduce the mass and counting measures and dimensions for subsets of Zd and explore the properties of large fractal subsets of the lattice. In particular, we prove analogues of Marstrand’s projection theorem and derive applications concerning the size of sumsets of dilated subsets of Z. We then show how to derive the classic projection theorems of Marstrand and Kaufman, along with an open conjecture of Oberlin, from their discrete analogues. In the third part, we study the existence of certain combinatorial configurations in members of a family of fractal subsets of the natural numbers, the Piatetski-Shapiro sets. We provide a metrical solution to the problem of determining which Piatetski-Shapiro sets contain infinitely many solutions to bivariate linear equations.
Vitaly Bergelson, Ph.D. (Advisor)
Hoi Nguyen, Ph.D. (Committee Member)
Nimish Shah, Ph.D. (Committee Member)
W. S. Winston Ho, Ph.D. (Other)
236 p.
Mathematics
Algebraic, analytic, and geometric notions of largeness

Recommended Citations

Citations

  • Glasscock, D. G. (2017). Algebraic, analytic, and geometric notions of largeness for subsets of Zd and their applications [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1499952794190004

    APA Style (7th edition)

  • Glasscock, Daniel. Algebraic, analytic, and geometric notions of largeness for subsets of Zd and their applications. 2017. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1499952794190004.

    MLA Style (8th edition)

  • Glasscock, Daniel. "Algebraic, analytic, and geometric notions of largeness for subsets of Zd and their applications." Doctoral dissertation, Ohio State University, 2017. http://rave.ohiolink.edu/etdc/view?acc_num=osu1499952794190004

    Chicago Manual of Style (17th edition)

osu1499952794190004
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© 2017, all rights reserved.
This open access ETD is published by The Ohio State University and OhioLINK.