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EG_Campolongo_Dissertation.pdf (1.94 MB)

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Lattice Point Counting through Fractal Geometry and Stationary Phase for Surfaces with Vanishing Curvature

Campolongo, Elizabeth Grace
http://orcid.org/0000-0003-0846-2413
http://rave.ohiolink.edu/etdc/view?acc_num=osu1658269573881902

Abstract Details

2022, Doctor of Philosophy, Ohio State University, Mathematics.
We explore lattice point counting and the method of stationary phase through the lens of questions about the number of lattice points on and near surfaces with vanishing curvature. Our focus is on spheres arising from the Heisenberg groups. In particular, we prove an upper bound on the number of points on and near large dilates of the unit spheres generated by the anisotropic Heisenberg norms for α ≥ 2. We accomplish this through a transformative process that takes a number theory question about counting lattice points and translates it into that of an analytical estimation of measure. This process relies on truncating and scaling the n-dimensional integer lattice to produce a fractal-like set. By introducing a measure on this resulting set and using elementary Fourier analysis, the counting problem is transformed into one of bounding an energy integral. This process uses principles of fractal geometry and oscillatory integrals. Primary challenges that arise are the presence of vanishing curvature and uneven dilations. Following a discussion and formal estimate of the curvature of the Heisenberg spheres, we utilize the method of stationary phase to compute a bound on the Fourier transform of their surface measures. Our work is inspired by that of Iosevich and Taylor (2011) and Garg, Nevo, and Taylor (2015). We present an extension of the main result in the former to surfaces with vanishing curvature. Furthermore, we utilize the techniques developed here to estimate the number of lattice points in the intersection of two such surfaces. Additionally, we present a mini-course on the basics of stationary phase—a quick-start guide to stationary phase in practice. This includes a discussion of the formulation of oscillatory integrals and their solutions with a focus on the impact of geometric properties (e.g. curvature) on the estimates for the decay of the Fourier transform. It further serves as a supplement to [Shakarchi and Stein, Functional Analysis: Chapter 8, Sections 2 and 3], providing detailed expositions of the proofs of fundamental results. We conclude with a discussion of potential avenues for improvement and interesting problems that could be considered utilizing our methods.
Krystal Taylor (Advisor)
Rodica Costin (Committee Member)
Barbara Keyfitz (Committee Member)
211 p.
Mathematics
Heisenberg norm; Heisenberg group; Heisenberg sphere; vanishing curvature; lattice point counting; stationary phase; oscillatory integrals; energy integrals; Hausdorff dimension; Fourier transform; surface measure; decay; fractal geometry; geometric measure theory; harmonic analysis; Fourier analysis; intersection of surfaces; Gauss circle problem

Recommended Citations

Citations

  • Campolongo, E. G. (2022). Lattice Point Counting through Fractal Geometry and Stationary Phase for Surfaces with Vanishing Curvature [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1658269573881902

    APA Style (7th edition)

  • Campolongo, Elizabeth. Lattice Point Counting through Fractal Geometry and Stationary Phase for Surfaces with Vanishing Curvature. 2022. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1658269573881902.

    MLA Style (8th edition)

  • Campolongo, Elizabeth. "Lattice Point Counting through Fractal Geometry and Stationary Phase for Surfaces with Vanishing Curvature." Doctoral dissertation, Ohio State University, 2022. http://rave.ohiolink.edu/etdc/view?acc_num=osu1658269573881902

    Chicago Manual of Style (17th edition)

osu1658269573881902
269
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This open access ETD is published by The Ohio State University and OhioLINK.