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Spatially Independent Martingales, Intersections, and Applications
 
Pablo Shmerkin Torcuato Di Tella University, Buenos Aires, Argentina and CONICET, Buenos Aires, Argentina
Ville Suomala University of Oulu, Oulu, Finland
Spatially Independent Martingales, Intersections, and Applications
Softcover ISBN:  978-1-4704-2688-0
Product Code:  MEMO/251/1195
List Price: $78.00
MAA Member Price: $70.20
AMS Member Price: $47.00
eBook ISBN:  978-1-4704-4264-4
Product Code:  MEMO/251/1195.E
List Price: $78.00
MAA Member Price: $70.20
AMS Member Price: $47.00
Softcover ISBN:  978-1-4704-2688-0
eBook: ISBN:  978-1-4704-4264-4
Product Code:  MEMO/251/1195.B
List Price: $156.00 $117.00
MAA Member Price: $140.40 $105.30
AMS Member Price: $94.00 $70.50
Spatially Independent Martingales, Intersections, and Applications
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Spatially Independent Martingales, Intersections, and Applications
Pablo Shmerkin Torcuato Di Tella University, Buenos Aires, Argentina and CONICET, Buenos Aires, Argentina
Ville Suomala University of Oulu, Oulu, Finland
Softcover ISBN:  978-1-4704-2688-0
Product Code:  MEMO/251/1195
List Price: $78.00
MAA Member Price: $70.20
AMS Member Price: $47.00
eBook ISBN:  978-1-4704-4264-4
Product Code:  MEMO/251/1195.E
List Price: $78.00
MAA Member Price: $70.20
AMS Member Price: $47.00
Softcover ISBN:  978-1-4704-2688-0
eBook ISBN:  978-1-4704-4264-4
Product Code:  MEMO/251/1195.B
List Price: $156.00 $117.00
MAA Member Price: $140.40 $105.30
AMS Member Price: $94.00 $70.50
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2512018; 102 pp
    MSC: Primary 28; 60; Secondary 42

    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 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 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.

  • Table of Contents
     
     
    • Chapters
    • 1. Introduction
    • 2. Notation
    • 3. The setting
    • 4. Hölder continuity of intersections
    • 5. Classes of spatially independent martingales
    • 6. A geometric criterion for Hölder continuity
    • 7. Affine intersections and projections
    • 8. Fractal boundaries and intersections with algebraic curves
    • 9. Intersections with self-similar sets and measures
    • 10. Dimension of projections:Applications of Theorem
    • 11. Upper bounds on dimensions of intersections
    • 12. Lower bounds for the dimension of intersections, and dimension conservation
    • 13. Products and convolutions of spatially independent martingales
    • 14. Applications to Fourier decay and restriction
  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2512018; 102 pp
MSC: Primary 28; 60; Secondary 42

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 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 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.

  • Chapters
  • 1. Introduction
  • 2. Notation
  • 3. The setting
  • 4. Hölder continuity of intersections
  • 5. Classes of spatially independent martingales
  • 6. A geometric criterion for Hölder continuity
  • 7. Affine intersections and projections
  • 8. Fractal boundaries and intersections with algebraic curves
  • 9. Intersections with self-similar sets and measures
  • 10. Dimension of projections:Applications of Theorem
  • 11. Upper bounds on dimensions of intersections
  • 12. Lower bounds for the dimension of intersections, and dimension conservation
  • 13. Products and convolutions of spatially independent martingales
  • 14. Applications to Fourier decay and restriction
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.