Spatially Independent Martingales, Intersections, and Applications.

Cover -- Title page -- Chapter 1. Introduction -- 1.1. Motivation and overview -- 1.2. General setup and major classes of examples -- Chapter 2. Notation -- Chapter 3. The setting -- 3.1. A class of random measures -- 3.2. Parametrized families of measures -- Chapter 4. Hölder continuity of intersections -- Chapter 5. Classes of spatially independent martingales -- 5.1. Cut-out measures arising from Poisson point processes -- 5.2. Subdivision fractals -- 5.3. Products of SI-martingales -- 5.4. Further examples -- Chapter 6. A geometric criterion for Hölder continuity -- Chapter 7. Affine intersections and projections -- 7.1. Main result on affine intersections -- 7.2. Applications: non-tube-null sets and Fourier decay -- Chapter 8. Fractal boundaries and intersections with algebraic curves -- 8.1. Intersections with algebraic curves -- 8.2. Polynomial projections -- Chapter 9. Intersections with self-similar sets and measures -- Chapter 10. Dimension of projections:Applications of Theorem 4.4 -- Chapter 11. Upper bounds on dimensions of intersections -- 11.1. Uniform upper bound for box dimension -- 11.2. Upper bounds for intersections with self-similar sets -- Chapter 12. Lower bounds for the dimension of intersections, and dimension conservation -- 12.1. Affine and algebraic intersections -- 12.2. Dimension conservation -- 12.3. Lower bounds on the dimension of intersections with self-similar sets -- Chapter 13. Products and convolutions of spatially independent martingales -- 13.1. Convolutions of random and deterministic measures -- 13.2. A generalization of Theorem 4.1 -- 13.3. Applications to cartesian products of measures and sets -- 13.4. Products and the breakdown of spatial independence -- 13.5. Self-products of SI-martingales -- Chapter 14. Applications to Fourier decay and restriction -- 14.1. Fourier decay of SI-martingales.

14.2. Application to the restriction problem for fractal measures -- Bibliography -- Back Cover.

The authors define a class of random measures, spatially independent martingales, which we view as a natural generalization of the canonical random discrete set, and which includes as special cases many variants of fractal percolation and Poissonian cut-outs. The authors pair the random measures with deterministic families of parametrized measures \{\eta_t\}_t, and show that under some natural checkable conditions, a.s. the mass of the intersections is H�lder continuous as a function of t. This continuity phenomenon turns out to underpin a large amount of geometric information about these measures, allowing us to unify and substantially generalize a large number of existing results on the geometry of random Cantor sets and measures, as well as obtaining many new ones. Among other things, for large classes of random fractals they establish (a) very strong versions of the Marstrand-Mattila projection and slicing results, as well as dimension conservation, (b) slicing results with respect to algebraic curves and self-similar sets, (c) smoothness of convolutions of measures, including self-convolutions, and nonempty interior for sumsets, and (d) rapid Fourier decay. Among other applications, the authors obtain an answer to a question of I. Łaba in connection to the restriction problem for fractal measures.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.