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Spatially independent martingales, intersections, and applications

About this Title

Pablo Shmerkin and Ville Suomala

Publication: Memoirs of the American Mathematical Society
Publication Year: 2018; Volume 251, Number 1195
ISBNs: 978-1-4704-2688-0 (print); 978-1-4704-4264-4 (online)
Published electronically: October 31, 2017
Keywords:Martingales, random measures, random sets, Hausdorff dimension, fractal percolation, random cutouts, convolutions, projections, intersections

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Table of Contents


  • Chapter 1. Introduction
  • Chapter 2. Notation
  • Chapter 3. The setting
  • Chapter 4. Hölder continuity of intersections
  • Chapter 5. Classes of spatially independent martingales
  • Chapter 6. A geometric criterion for Hölder continuity
  • Chapter 7. Affine intersections and projections
  • Chapter 8. Fractal boundaries and intersections with algebraic curves
  • 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
  • Chapter 12. Lower bounds for the dimension of intersections, and dimension conservation
  • Chapter 13. Products and convolutions of spatially independent martingales
  • Chapter 14. Applications to Fourier decay and restriction


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 , and show that under some natural checkable conditions, a.s. the mass of the intersections is Hölder continuous as a function of . 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. Łaba in connection to the restriction problem for fractal measures.

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