Skip to main content

Recent Progress on Dimensions of Projections

  • Conference paper
  • First Online:
Geometry and Analysis of Fractals

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 88))

Abstract

This is a survey on recent progress on the question: how do projections effect dimensions generically? I shall also discuss briefly dimensions of plane sections.

The author was supported by the Academy of Finland.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Balogh, Z.M., Cartagena, D.E., Fässler, K., Mattila, P., Tyson, J.T.: The effect of projections on dimension in the Heisenberg group, Revista Mat. Iberoamericana 29, 381–432 (2013)

    Google Scholar 

  2. Balogh, Z.M., Fässler, K., Mattila, P., Tyson, J.T.: Projection and slicing theorems in Heisenberg groups. Adv. Math. 231, 569–604 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  3. Balogh, Z.M., Tyson, J.T., Warhurst, B.: Sub-Riemannian versus Euclidean dimension comparision and fractal geometry on Carnot groups. Adv. Math. 220, 560–619 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  4. Barany, B., Ferguson, A., Simon, K.: Slicing the Sierpinski gasket. Nonlinearity 25, 1753–1770 (2012)

    Google Scholar 

  5. Bourgain, J.: Discretized sum-product and projection theorems. J. Anal. Math. 112, 193–236 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  6. Falconer, K.J.: Hausdorff dimension and the exceptional set of projections. Mathematika 29, 90–96 (1982)

    Article  MathSciNet  Google Scholar 

  7. Falconer, K.J.: Geometry of Fractal Sets. Cambridge University Press, Cambridge (1985)

    Google Scholar 

  8. Falconer, K.J., Howroyd, J.D.: Projection theorems for box and packing dimensions. Math. Proc. Camb. Phil. Soc. 119, 287–295 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  9. Falconer, K.J., Howroyd, J.D.: Packing dimension of projections and dimension profiles. Math. Proc. Camb. Phil. Soc. 121, 269–286 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  10. Falconer, K.J., Mattila, P.: Packing dimension of projections and sections of measures. Math. Proc. Camb. Phil. Soc. 119, 695–713 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  11. Fässler, K., Orponen, T.: Constancy results for special families of projections. Math. Proc. Camb. Philos. Soc. 154, 549–568 (2013). arXiv:1208.1876

  12. Fässler, K., Orponen, T.: On restricted families of projections in \({\mathbb{R}}^3\). arXiv:1302.6550

  13. Federer, H.: Geometric Measure Theory. Springer, New York (1969)

    Google Scholar 

  14. Fraser, J., Orponen, T., Sahlsten, T.: On Fourier analytic properties of graphs. IMRN. arXiv:1212.4813 (to appear)

  15. Furstenberg, H.: Ergodic fractal measures and dimension conservation. Ergodic Theo. Dynam. Syst. 28, 405–422 (2008)

    MATH  MathSciNet  Google Scholar 

  16. Hochman, M.: On self-similar sets with overlaps and inverse theorems for entropy. arXiv:1212.1873

  17. Hochman, M., Shmerkin, P.: Local entropy averages and projections of fractal measures. Ann. Math. 175, 1001–1059 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  18. Hovila, R.: Transversality of isotropic projections, unrectifiability and Heisenberg groups. Revista Mat. Iberoamericana (to appear)

    Google Scholar 

  19. Hovila, R., Järvenpää, E., Järvenpää, M., Ledrappier, F.: Besicovitch-Federer projection theorem and geodesic flows on Riemann surfaces. Geom. Dedicata. 161, 51–61 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  20. Hovila, R., Järvenpää, E., Järvenpää, M., Ledrappier, F.: Singularity of projections of 2-dimensional measures invariant under the geodesic flow. Comm. Math. Phys. 312, 127–136 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  21. Järvenpää, E., Järvenpää, M., Keleti, T.: Hausdorff dimension and non-degenerate families of projections. J. Geom. Anal. (2013)

    Google Scholar 

  22. Järvenpää, E., Järvenpää, M., Ledrappier, F., Leikas, M.: (Non)regularity of projections of measures invariant under geodesic flow. Comm. Math. Phys. 254, 695–717 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  23. Järvenpää, E., Järvenpää, M., Leikas, M.: One-dimensional families of projections. Nonlinearity 21, 453–463 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  24. Järvenpää, M.: On the upper Minkowski dimension, packing dimension, and orthogonal projections. Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 99, 34 (1994)

    Google Scholar 

  25. Kaufman, R.: On Hausdorff dimension of projections. Mathematika 15, 153–155 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  26. Kaufman, R., Mattila, P.: Hausdorff dimension and exceptional sets of linear transformations. Ann. Acad. Sci. Fenn. Ser. A I Math. 1, 387–392 (1975)

    Google Scholar 

  27. Liu, Q.H., Xi, L.F., Zhao, Y.F.: Dimensions of intersections of the Sierpinski carpet with lines of rational slopes. Proc. Edinb. Math. Soc. 50, 411–428 (2007)

    Google Scholar 

  28. Manning, A., Simon, K.: Dimension of slices through the Sierpinski carpet. Trans. Amer. Math. Soc. 365, 213–250 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  29. Marstrand, J.M.: Some fundamental geometrical properties of plane sets of fractional dimensions. Proc. London Math. Soc. 3(4), 257–302 (1954)

    Google Scholar 

  30. Mattila, P.: Hausdorff dimension, orthogonal projections and intersections with planes. Ann. Acad. Sci. Fenn. Ser. A I Math. 1, 227–244 (1975)

    Google Scholar 

  31. Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press, Cambridge (1995)

    Google Scholar 

  32. Mattila, P.: Hausdorff dimension, projections, and the Fourier transform. Publ. Mat. 46, 3–48 (2004)

    Article  MathSciNet  Google Scholar 

  33. Mattila, P.: Marstrand’s theorems. In: Córdoba, A., Férnandez, J.L., Férnandez, P. (eds.) All That Math, Portraits of Mathematicians as Young Readers, pp. 235–245. Revista Mathematica Iberoamericana (2011)

    Google Scholar 

  34. Oberlin, D.M.: Restricted Radon transforms and projections of planar set. Canad. Math. Bull. 55, 815–820 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  35. Oberlin, D.M.: Exceptional sets of projections, unions of k-planes, and associated transforms. arXiv:1107.4913

  36. Orponen, T.: Slicing sets and measures, and the dimension of exceptional parameters. J. Geom. Anal. 24, 47–80 (2012)

    Google Scholar 

  37. Orponen, T.: On the packing dimension and category of exceptional sets of orthogonal projections. arXiv:1204.2121

  38. Orponen, T.: Hausdorff dimension estimates for some restricted families of projections in \({\mathbb{R}}^3\). arXiv:1304.4955

  39. Peres, Y., Schlag, W.: Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions. Duke Math. J. 102, 193–251 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  40. Peres, Y., Shmerkin, P.: Resonance between Cantor sets. Ergodic Theor. Dynam. Syst. 29, 201–221 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  41. Rams, M.: Packing dimension estimation for exceptional parameters. Israel J. Math. 130, 125–144 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  42. Wen, Z., Wu, W., Xi, L.: Dimension of slices through a self-similar set with initial cubical pattern. Ann. Acad. Sci. Fenn. Ser. A I (to appear)

    Google Scholar 

  43. Wen, Z.-Y., Xi, L.: On the dimension of sections for the graph-directed sets. Ann. Acad. Sci. Fenn. Ser. A I(35), 515–535 (2010)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Pertti Mattila .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Mattila, P. (2014). Recent Progress on Dimensions of Projections. In: Feng, DJ., Lau, KS. (eds) Geometry and Analysis of Fractals. Springer Proceedings in Mathematics & Statistics, vol 88. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43920-3_10

Download citation

Publish with us

Policies and ethics