University of Oulu

Shmerkin, P., Suomala, V. (2018) Spatially independent martingales, intersections, and applications. Memoirs of the American Mathematical Society, 251, 1195.

Spatially independent martingales, intersections, and applications

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Author: Shmerkin, Pablo1; Suomala, Ville2
Organizations: 1Department of Mathematics and Statistics, Torcuato Di Tella University, and CONICET, Buenos Aires, Argentina
2Department of Mathematical Sciences, University of Oulu, Finland
Format: article
Version: accepted version
Access: open
Online Access: PDF Full Text (PDF, 1.4 MB)
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Language: English
Published: American Mathematical Society, 2018
Publish Date: 2019-01-14


We 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. We pair the random measures with deterministic families of parametrized measures {ηt}t, and show that under some natural checkable conditions, a.s. the mass of the intersections is Hölder 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 we 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, (d) rapid Fourier decay. Among other applications, we obtain an answer to a question of I. Laba in connection to the restriction problem for fractal measures.

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Series: Memoirs of the American Mathematical Society
ISSN: 0065-9266
ISSN-E: 1947-6221
ISSN-L: 0065-9266
ISBN: 978-1-4704-2688-0
ISBN Print: 978-1-4704-4264-4
Volume: 251
Article number: 1195
DOI: 10.1090/memo/1195
Type of Publication: A1 Journal article – refereed
Field of Science: 111 Mathematics
Copyright information: © 2017 Memoirs of the American Mathematical Society. First published in Memoirs of the American Mathematical Society in Year: 2018; Volume 251, Number 1195 published by the American Mathematical Society,