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Arithmetic Structures in Small Subsets of Euclidean Space

Carnovale, Marc
http://rave.ohiolink.edu/etdc/view?acc_num=osu1555657038785892

Abstract Details

2019, Doctor of Philosophy, Ohio State University, Mathematics.
In this thesis we extend techniques from additive combinatorics to the setting of harmonic analysis and geometric measure theory. We focus on studying the distribution of three-term arithmetic progressions (3APs) within the supports of singular measures in Euclidean space. In Chapter 2 we prove a relativized version of Roth's theorem on existence of 3APs for positive measure subsets of pseudorandom measures on R and show that a positive measure of points are the basepoints for three-term arithmetic progressions within these measures' supports. In Chapter 3 we combine Mattila's approach to the Falconer distance conjecture with Green and Tao's arithmetic regularity lemma to show that measures on R^d with sufficiently small Fourier transform as measured by an L^p-norm have supports with an abundance of three-term arithmetic progressions of various step-sizes. In Chapter 4 we develop a novel regularity lemma to show that measures on R^d with sufficiently large dimension, as measured by a gauge function, must either contain non-trivial three-term arithmetic progressions in their supports or else be structured in a specific quantitative manner, which can be qualitatively described as, at infinitely many scales, placing a large amount of mass on at least two distinct cosets of a long arithmetic progression.
Vitaly Bergelson (Advisor)
Alexander Leibman (Committee Member)
Krystal Taylor (Committee Member)
112 p.
Mathematics
Fourier analysis; additive combinatorics; geometric measure theory; arithmetic progressions; fractals; bohr sets; roths theorem

Recommended Citations

Citations

  • Carnovale, M. (2019). Arithmetic Structures in Small Subsets of Euclidean Space [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1555657038785892

    APA Style (7th edition)

  • Carnovale, Marc. Arithmetic Structures in Small Subsets of Euclidean Space. 2019. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1555657038785892.

    MLA Style (8th edition)

  • Carnovale, Marc. "Arithmetic Structures in Small Subsets of Euclidean Space." Doctoral dissertation, Ohio State University, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=osu1555657038785892

    Chicago Manual of Style (17th edition)

osu1555657038785892
462
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This open access ETD is published by The Ohio State University and OhioLINK.