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Hausdorff Dimension, Projections, Intersections, and Besicovitch Sets

  • Pertti MattilaEmail author
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

This is a survey on recent developments on the Hausdorff dimension of projections and intersections for general subsets of Euclidean spaces, with an emphasis on estimates of the Hausdorff dimension of exceptional sets and on restricted projection families. We shall also discuss relations between projections and Hausdorff dimension of Besicovitch sets.

Keywords

Hausdorff dimension Orthogonal projection Intersection Besicovitch set 

Subject Classification:

28A75 

References

  1. 1.
    Z.M. Balogh, E. Durand-Cartagena, K. Fässler, P. Mattila, J.T. Tyson, The effect of projections on dimension in the Heisenberg group. Rev. Mat. Iberoam. 29, 381–432 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Z.M. Balogh, K. Fässler, P. Mattila, J.T. Tyson, Projection and slicing theorems in Heisenberg groups. Adv. Math. 231, 569–604 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Z.M. Balogh, A. Iseli, Dimension distortion by projections on Riemannian surfaces of constant curvature. Proc. Amer. Math. Soc. 144, 2939–2951 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Z.M. Balogh, A. Iseli. Marstrand type projection theorems for normed spaces, to appear in J. Fractal GeomGoogle Scholar
  5. 5.
    V. Beresnevich, K.J. Falconer, S. Velani, A. Zafeiropoulos, Marstrand’s Theorem Revisited: Projecting Sets of Dimension Zero, arXiv:1703.08554
  6. 6.
    A.S. Besicovitch, On fundamental geometric properties of plane-line sets. J. London Math. Soc. 39, 441–448 (1964)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    J. Bourgain, Besicovitch type maximal operators and applications to Fourier analysis. Geom. Funct. Anal. 1, 147–187 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    J. Bourgain, On the dimension of Kakeya sets and related maximal inequalities. Geom. Funct. Anal. 9, 256–282 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    J. Bourgain, On the Erdös-Volkmann and Katz-Tao ring conjectures. Geom. Funct. Anal. 13, 334–365 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    J. Bourgain, The discretized sum-product and projection theorems. J. Anal. Math. 112, 193–236 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    C. Chen, Projections in vector spaces over finite fields. Ann. Acad. Sci. Fenn. A Math. 43, 171–185 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    C. Chen, Restricted families of projections and random subspaces, arXiv:1706.03456
  13. 13.
    R.O. Davies, Some remarks on the Kakeya problem. Proc. Cambridge Philos. Soc. 69, 417–421 (1971)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Z. Dvir, On the size of Kakeya sets in finite fields. J. Amer. Math. Soc. 22, 1093–1097 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    M.B. Erdoğan, A bilinear Fourier extension problem and applications to the distance set problem. Int. Math. Res. Not. 23, 1411–1425 (2005)zbMATHCrossRefGoogle Scholar
  16. 16.
    K.J. Falconer, Sections of sets of zero Lebesgue measure. Mathematika 27, 90–96 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    K.J. Falconer, Hausdorff dimension and the exceptional set of projections. Mathematika 29, 109–115 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    K.J. Falconer, The Geometry of Fractal Sets (Cambridge University Press, Cambridge, 1985)zbMATHCrossRefGoogle Scholar
  19. 19.
    K.J. Falconer, Classes of sets with large intersection. Mathematika 32, 191–205 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    K.J. Falconer, On the Hausdorff dimension of distance sets. Mathematika 32, 206–212 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    K.J. Falconer, J. Fraser, X. Jin, Sixty Years of Fractal Projections. Fractal geometry and stochastics V, 3–25, Progr. Probab., 70, Birkhäuser/Springer, Cham (2015)zbMATHCrossRefGoogle Scholar
  22. 22.
    K.J. Falconer, P. Mattila, Strong Marstrand theorems and dimensions of sets formed by subsets of hyperplanes. J. Fractal Geom. 3, 319–329 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    K.J. Falconer, T. O’Neil, Convolutions and the geometry of multifractal measures. Math. Nachr. 204, 61–82 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    K. Fässler, R. Hovila, Improved Hausdorff dimension estimate for vertical projections in the Heisenberg group. Ann. Sc. Norm. Super. Pisa Cl. Sci. 15(5), 459–483 (2016)Google Scholar
  25. 25.
    K. Fässler, T. Orponen, On restricted families of projections in \({\mathbb{R}}^3\). Proc. London Math. Soc. 109(3), 353–381 (2014)Google Scholar
  26. 26.
    H. Federer, Geometric Measure Theory (Springer, 1969)Google Scholar
  27. 27.
    L. Guth, Polynomial Methods in Combinatorics (American Mathematical Society, Provedence, RI, 2016)zbMATHCrossRefGoogle Scholar
  28. 28.
    W. He, Orthogonal projections of discretized sets, arXiv:1710.00759, to appear in J. Fractal Geom
  29. 29.
    K. Héra, T. Keleti, A. Máthé. Hausdorff dimension of union of affine subspaces, arXiv:1701.02299, to appear in J. Fractal Geom
  30. 30.
    R. Hovila, E. Järvenpää, M. Järvenpää, F. Ledrappier, Besicovitch-Federer projection theorem and geodesic flows on Riemann surfaces. Geom. Dedicata 161, 51–61 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    R. Hovila, E. Järvenpää, M. Järvenpää, F. Ledrappier, Singularity of projections of 2-dimensional measures invariant under the geodesic flows on Riemann surfaces. Comm. Math. Phys. 312, 127–136 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    A. Iosevich, B. Liu, Falconer distance problem, additive energy and Cartesian products. Ann. Acad. Sci. Fenn. Math. 41, 579–585 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    A. Iosevich, B. Liu, Pinned distance problem, slicing measures and local smoothing estimates, arXiv:1706.09851
  34. 34.
    E. Järvenpää, M. Järvenpää, T. Keleti, Hausdorff dimension and non-degenerate families of projections. J. Geom. Anal. 24, 2020–2034 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    E. Järvenpää, M. Järvenpää, F. Ledrappier, M. Leikas, One-dimensional families of projections. Nonlinearity 21(3), 453–463 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    A. Käenmäki, T. Orponen, L. Venieri, A Marstrand-type restricted projection theorem in \({\mathbb{R}}^3\), arXiv:1708.04859
  37. 37.
    J.-P. Kahane, Sur la dimension des intersections, In Aspects of Mathematics and Applications. North-Holland Math. Lib. 34, 419–430 (1986)CrossRefGoogle Scholar
  38. 38.
    N.H. Katz, T. Tao, Bounds on arithmetic projections, and applications to the Kakeya conjecture. Math. Res. Lett. 6, 625–630 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    N.H. Katz, T. Tao, New bounds for Kakeya problems. J. Anal. Math. 87, 231–263 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    N.H. Katz, J. Zahl, An improved bound on the Hausdorff dimension of Besicovitch sets in \({\mathbb{R}}^3\). J. Amer. Math. Soc., August 2018, published electronicallyGoogle Scholar
  41. 41.
    R. Kaufman, On Hausdorff dimension of projections. Mathematika 15, 153–155 (1968)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    R. Kaufman, P. Mattila, Hausdorff dimension and exceptional sets of linear transformations. Ann. Acad. Sci. Fenn. A Math. 1, 387–392 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    T. Keleti. A 1-dimensional subset of the reals that intersects each of its translates in at most a single point. Real Anal. Exchange 24, 843–844 (1998/99)Google Scholar
  44. 44.
    T. Keleti, Are lines much bigger than line segments? Proc. Amer. Math. Soc. 144, 1535–1541 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    T. Keleti, Small union with large set of centers, in Recent developments in Fractals and Related Fields, ed. Julien Barral and Stéphane Seuret, Birkhäuser (2015). arXiv:1701.02762
  46. 46.
    R. Lucá, K. Rogers, Average decay of the Fourier transform of measures with applications, arXiv:1503.00105
  47. 47.
    J.M. Marstrand, Some fundamental geometrical properties of plane sets of fractional dimensions. Proc. London Math. Soc. 4(3), 257–302 (1954)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    J.M. Marstrand, The dimension of cartesian product sets. Proc. Cambridge Philos. Soc. 50(3), 198–202 (1954)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    J.M. Marstrand, Packing planes in \({{\mathbb{R}}}^3\). Mathematika 26, 180–183 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    P. Mattila, Hausdorff dimension, orthogonal projections and intersections with planes. Ann. Acad. Sci. Fenn. A Math. 1, 227–244 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    P. Mattila, Hausdorff dimension and capacities of intersections of sets in n-space. Acta Math. 152, 77–105 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    P. Mattila, On the Hausdorff dimension and capacities of intersections. Mathematika 32, 213–217 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    P. Mattila, Geometry of Sets and Measures in Euclidean Spaces (Cambridge University Press, Cambridge, 1995)zbMATHCrossRefGoogle Scholar
  54. 54.
    P. Mattila, Recent Progress on Dimensions of Projections. Geometry and analysis of fractals, 283–301, Springer Proc. Math. Stat., 88, Springer, Heidelberg (2014)Google Scholar
  55. 55.
    P. Mattila, Fourier Analysis and Hausdorff Dimension (Cambridge University Press, Cambridge, 2015)zbMATHCrossRefGoogle Scholar
  56. 56.
    P. Mattila, Exeptional set estimates for the Hausdorff dimension of intersections. Ann. Acad. Sci. Fenn. A Math. 42, 611–620 (2017)zbMATHCrossRefGoogle Scholar
  57. 57.
    P. Mattila, T. Orponen, Hausdorff dimension, intersections of projections and exceptional plane sections. Proc. Amer. Math. Soc. 144, 3419–3430 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    P. Mattila, P. Sjölin, Regularity of distance measures and sets. Math. Nachr. 204, 157–162 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    U. Molter, E. Rela, Improving dimension estimates for Furstenberg-type sets. Adv. Math. 223, 672–688 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    U. Molter, E. Rela, Furstenberg sets for a fractal set of directions. Proc. Amer. Math. Soc. 140, 2753–2765 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    U. Molter, E. Rela, Small Furstenberg sets. J. Math. Anal. Appl. 400, 475–486 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    D.M. Oberlin, Restricted Radon transforms and projections of planar sets. Canad. Math. Bull. 55, 815–820 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    D.M. Oberlin, Exceptional sets of projections, unions of k-planes, and associated transforms. Israel J. Math. 202, 331–342 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    D.M. Oberlin, Some toy Furstenberg sets and projections of the four-corner Cantor set. Proc. Amer. Math. Soc. 142(4), 1209–1215MathSciNetzbMATHCrossRefGoogle Scholar
  65. 65.
    D.M. Oberlin, R. Oberlin, Application of a Fourier restriction theorem to certain families of projections in \({{\mathbb{R}}}^3\). J. Geom. Anal. 25(3), 1476–1491 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  66. 66.
    R. Oberlin, Two bounds for the \(X\)-ray transform. Math. Z. 266, 623–644 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  67. 67.
    T. Orponen, Slicing sets and measures, and the dimension of exceptional parameters. J. Geom. Anal. 24, 47–80 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  68. 68.
    T. Orponen, Hausdorff dimension estimates for some restricted families of projections in \({\mathbb{R}}^3\). Adv. Math. 275, 147–183 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  69. 69.
    T. Orponen, On the packing dimension and category of exceptional sets of orthogonal projections. Ann. Mat. Pura Appl. 194(4), 843–880 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  70. 70.
    T. Orponen, Projections of planar sets in well-separated directions. Adv. Math. 297, 1–25 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  71. 71.
    T. Orponen, On the distance sets of AD-regular sets. Adv. Math. 307, 1029–1045 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  72. 72.
    T. Orponen, A sharp exceptional set estimate for visibility. Bull. London. Math Soc. 50, 1–6 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  73. 73.
    T. Orponen, An improved bound on the packing dimension of Furstenberg sets in the plane, arXiv:1611.09762, to appear in J. Eur. Math. Soc
  74. 74.
    T. Orponen, On the dimension and smoothness of radial projections, arXiv:1710.11053, to appear in Anal. PDE
  75. 75.
    T. Orponen, L. Venieri, Improved bounds for restricted families of projections to planes in \({\mathbb{R}}^3\), appeared online in Int (Math. Res, Notices, 2018)Google Scholar
  76. 76.
    Y. Peres, W. Schlag, Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions. Duke Math. J. 102, 193–251 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  77. 77.
    E. Rela, Refined size estimates for Furstenberg sets via Hausdorff measures: a survey of some recent results, arXiv:1305.3752
  78. 78.
    P. Shmerkin, Projections of self-similar and related fractals: a survey of recent developments, arXiv:1501.00875, Fractal geometry and stochastics V, 53–74, Progr. Probab., 70, Birkhäuser/Springer, Cham (2015)zbMATHCrossRefGoogle Scholar
  79. 79.
    P. Shmerkin, On Furstenberg’s intersection conjecture, self-similar measures, and the \(L^q\) norms of convolutions, arXiv:1609.07802
  80. 80.
    P. Shmerkin, On distance sets, box-counting and Ahlfors-regular sets. Discrete Anal. Paper No. 9, 22 pp (2017)Google Scholar
  81. 81.
    P. Shmerkin, On the Hausdorff dimension of pinned distance sets, arXiv:1706.00131
  82. 82.
    L. Venieri, Dimension estimates for Kakeya sets defined in an axiomatic setting, arXiv:1703.03635, Ann. Acad. Sci Fenn. Disserationes 161 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  83. 83.
    T.W. Wolff, An improved bound for Kakeya type maximal functions. Rev. Mat. Iberoam. 11, 651–674 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  84. 84.
    T.W. Wolff, A Kakeya-type problem for circles. Amer. J. Math. 119, 985–1026 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  85. 85.
    T.W. Wolff, Decay of circular means of Fourier transforms of measures. Int. Math. Res. Not. 10, 547–567 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  86. 86.
    T.W. Wolff, Lectures on Harmonic Analysis, vol. 29 (American Mathematical Society, 2003)Google Scholar
  87. 87.
    M. Wu, A proof of Furstenberg’s conjecture on the intersections of \(xp\) and \(xq\) invariant sets, arXiv:1609.08053
  88. 88.
    H. Yu, Kakeya books and projections of Kakeya sets, arXiv:1704.04488

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Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of HelsinkiHelsinkiFinland

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