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Zhu, Junjie

University of British Columbia

Harmonic analysis studies functions and sets through their frequencies. In harmonic analysis, central questions such as the restriction, Kakeya, and Bochner-Riesz conjectures involve oscillatory integrals. Geometric measure theory is a field that analyzes the geometric properties of sets in Euclidean spaces through measures supported on them. This dissertation concerns two problems at the intersection of harmonic analysis and geometric measure theory, with the common theme of applying oscillatory integral methods. The first problem involves notions of the dimensions of sets. The dimension is a key concept in geometric measure theory, which aims to capture the size of a set. The problem under consideration uses two distinct notions, Hausdorff and Fourier dimensions, in the context of hypersurfaces. Any hypersurface in Rᵈ⁺¹ has a Hausdorff dimension of d. However, the Fourier dimension depends on finer geometric properties of the hypersurface, such as curvature. For example, the Fourier dimension of a hyperplane is 0, and that of a hypersurface with non-vanishing Gaussian curvature is d. In 2022, Fraser, Harris, and Kroon showed that the Euclidean light cone in RRᵈ⁺¹ has a Fourier dimension of d-1, which leads one to conjecture that the Fourier dimension of a hypersurface equals the number of non-vanishing principal curvatures. This thesis proves this conjecture for all constant rank hypersurfaces. The method involves substantial generalizations of their strategy. The second problem involves recognizing quadratic patterns in R. Many results in harmonic analysis and geometric measure theory ensure the existence of geometric configurations, such as arithmetic progressions and vertices of triangles, under the largeness of sets. In a seminal work by Laba and Pramanik in 2009, the largeness of a set was quantified by two measure-theoretic criteria, one involving the mass on Euclidean balls and the other using the decay properties of the Fourier transform of the measure. Recently, Kuca, Orponen, Sahlsten, and also Bruce, Pramanik removed the Fourier decay condition and showed that arbitrary sets with large Hausdorff dimensions contain two-point non-linear patterns, such as (x,y) and (x+t,y+t²). This thesis explores the existence of a three-point non-linear pattern x,x+t,x+t² in sets of large Hausdorff dimensions.

Text

2025-04-17

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/90687

Mathematics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nc-nd/4.0/