Spatially independent Martingales, intersections, and applications
Author(s)
Bibliographic Information
Spatially independent Martingales, intersections, and applications
（Memoirs of the American Mathematical Society, no. 1195）
American Mathematical Society, c2017
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Note
Includes bibliographical references (p. 99-102)
January 2018, volume 251, number 1195 (second of 6 numbers)
Description and Table of Contents
Description
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 Holder 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. Laba in connection to the restriction problem for fractal measures.
Table of Contents
Introduction
Notation
The setting
Holder continuity of intersections
Classes of spatially independent martingales
A geometric criterion for Holder continuity
Affine intersections and projections
Fractal boundaries and intersections with algebraic curves
Intersections with self-similar sets and measures
Dimension of projections: applications of Theorem 4.4
Upper bounds on dimensions of intersections
Lower bounds for the dimension of intersections, and dimension conservation
Products and convolutions of spatially independent martingales
Applications to Fourier decay and restriction
Bibliography
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