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 Laba in connection to the restriction problem for fractal measures

Spatially Independent Martingales, Intersections, and Applications (Memoirs of the American Mathematical Society, 1195) PDF

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