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Spatially Independent Martingales Intersections and Applications (Memoirs of the American Mathematical Society)

Probability & Statistics

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About the Author Pablo Shmerkin, Torcuato Di Tella University, Buenos Aires, Argentina and CONICET, Buenos Aires, Argentina.Ville Suomala, University of Oulu, Finland. Product 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.

Product Specifications

Format: paperback
ASIN: 1470426889
Category: Books > Subjects > Science, Nature & Maths > Mathematics > Probability & Statistics
Domain: Amazon UK
Release Date: 30 March 2018
Listed Since: 25 October 2017

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Spatially Independent Martingales Intersections and Applications (Memoirs of the American Mathematical Society)

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