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Spatially Independent Martingales Intersections and Applications 1st Edition by Pablo Shmerkin, Ville Suomala ISBN 9781470442644 1470442647

Name: Spatially Independent Martingales Intersections and Applications 1st Edition by Pablo Shmerkin, Ville Suomala ISBN 9781470442644 1470442647
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Spatially Independent Martingales Intersections and Applications 1st Edition Pablo Shmerkin Digital Instant Download

Author(s): Pablo Shmerkin; Ville Suomala

ISBN(s): 9781470442644, 1470442647

Edition: 1

File Details: PDF, 1.76 MB

Year: 2017

Language: English

SKU: EB-51629896 Category: Uncategorized Tags: Pablo Shmerkin, Ville Suomala

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Spatially Independent Martingales Intersections and Applications 1st Edition by Pablo Shmerkin, Ville Suomala – Ebook PDF Instant Download/Delivery: 9781470442644 ,1470442647
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ISBN 10: 1470442647
ISBN 13: 9781470442644
Author: Pablo Shmerkin, Ville Suomala

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.

Spatially Independent Martingales Intersections and Applications 1st Edition Table of contents:

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

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Tags: Pablo Shmerkin, Ville Suomala, Spatially Independent, Applications