Discrepancy and energy in various geometric settings
2023-12
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Discrepancy and energy in various geometric settings
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2023-12
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This dissertation details the author’s work on several related problems in the study of irregularities of distribution, particularly the geometric discrepancy. We begin by surveying the history of the field starting with the basic notion of equidistributed sequences in one dimension. One may quantify the distribution properties of a sequence—or more generally a point set—in some space via the discrepancy, which measures the deviation of the set from an ideal distribution. We discuss a number of important results and open problems in the study of the discrepancy function with respect to different geometric classes of sets on both the torus and the sphere, drawing connections to harmonic analysis. A significant part of the dissertation is devoted to the problem of determining the correct asymptotic for the discrepancy with respect to various classes of rotated rectangles in two dimensions. The two “extreme” cases—when the underlying class of sets consists of axis-parallel rectangles (a single direction) vs. arbitrarily rotated rectangles (all directions)—have been well-studied: the discrepancy in the former case is logarithmic in the number of points, while in the latter case it is polynomial. Thus, there is a natural question of what happens in various intermediate cases. We make partial progress on this problem in particular by proving a lower bound on the directional discrepancy where the allowed set of directions is a restricted interval. We then turn to some related problems regarding the discrepancy on the sphere. We give an overview of Beck’s bounds for the spherical cap discrepancy (which are akin to those for arbitrarily rotated rectangles in the plane), and then prove a refinement of Beck’s lower bound which removes a layer of averaging. We also consider upper bounds for the spherical cap discrepancy, and in particular give a new example of a point set (the HEALPix point set) that achieves the current best-known asymptotic for an upper bound on the spherical cap discrepancy for deterministic point sets. We conclude the paper with a brief discussion of a “greedy” sequence on the sphere which has good distribution properties in the sense of the L2-discrepancy and for which numerics suggest it may in fact achieve close to optimal discrepancy.
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University of Minnesota Ph.D. dissertation. December 2023. Major: Mathematics. Advisor: Max Engelstein. 1 computer file (PDF); iii, 135 pages.
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Mastrianni, Michelle. (2023). Discrepancy and energy in various geometric settings. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/260653.
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