Skip to main content
SpringerLink
Log in

A Class of Random Cantor Measures, with Applications

  • Conference paper
  • First Online:
Recent Developments in Fractals and Related Fields (FARF3 2015)

Part of the book series: Trends in Mathematics ((TM))

Included in the following conference series:

Abstract

We survey some of our recent results on the geometry of spatially independent martingales, in a more concrete setting that allows for shorter, direct proofs, yet is general enough for several applications and contains the well-known fractal percolation measure. We study self-convolutions and Fourier decay of measures in our class, and present applications of these results to the restriction problem for fractal measures, and the connection between arithmetic structure and Fourier decay.

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

Access this chapter

Log in via an institution
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
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
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

Notes

  1. 1.

    After this work was completed, Łaba and Wang [20] found the optimal relationship between dimension and restriction, up to the endpoint.

References

  1. Bak, J.-G., Seeger, A.: Extensions of the Stein-Tomas theorem. Math. Res. Lett. 18(4), 767–781 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Barral, J., Mandelbrot, B.B.: Multifractal products of cylindrical pulses. Probab. Theory Relat. Fields 124(3), 409–430 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Behrend, F.A.: On sets of integers which contain no three terms in arithmetical progression. Proc. Nat. Acad. Sci. USA 32, 331–332 (1946)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen, X.: A Fourier restriction theorem based on convolution powers. Proc. Am. Math. Soc. 142(11), 3897–3901 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, X.: Sets of Salem type and sharpness of the L 2-Fourier restriction theorem. Trans. Am. Math. Soc. 368(3), 1959–1977 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, X., Seeger, A.: Convolution powers of Salem measures with applications. Can. J. Math. 69(2), 284–320 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  7. Davies, R.O., Marstrand, J.M., Taylor, S.J.: On the intersections of transforms of linear sets. Colloq. Math. 7, 237–243 (1959/1960)

    MathSciNet  Google Scholar 

  8. Ekström, F., Persson, T., Schmeling, J.: On the Fourier dimension and a modification. J. Fractal Geom. 2(3), 309–337 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1992)

    MATH  Google Scholar 

  10. Hambrook, K., Łaba, I.: On the sharpness of Mockenhaupt’s restriction theorem. Geom. Funct. Anal. 23(4), 1262–1277 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Henriot, K., Łaba, I., Pramanik, M.: On polynomial configurations in fractal sets. Anal. PDE 9(5), 1153–1184 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Janson, S.: Large deviations for sums of partly dependent random variables. Random Struct. Algoritm. 24(3), 234–248 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kahane, J.-P.: Some Random Series of Functions. Cambridge Studies in Advanced Mathematics, vol. 5, 2nd edn. Cambridge University Press, Cambridge (1985)

    Google Scholar 

  14. Kahane, J.-P.: Positive martingales and random measures. Chin. Ann. Math. B 8(1), 1–12 (1987). A Chinese summary appears in Chinese Ann. Math. Ser. A 8(1), 136 (1987)

    Google Scholar 

  15. Kaufman, R.: On the theorem of Jarník and Besicovitch. Acta Arith. 39(3), 265–267 (1981)

    MathSciNet  MATH  Google Scholar 

  16. Keleti, T.: A 1-dimensional subset of the reals that intersects each of its translates in at most a single point. Real Anal. Exch. 24(2):843–844 (1998/1999)

    MathSciNet  MATH  Google Scholar 

  17. Körner, T.: On a theorem of Saeki concerning convolution squares of singular measures. Bull. Soc. Math. France 136(3), 439–464 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Łaba, I.: Harmonic analysis and the geometry of fractals. In: Proceedings of the 2014 International Congress of Mathematicians (to appear, 2014)

    Google Scholar 

  19. Łaba, I., Pramanik, M.: Arithmetic progressions in sets of fractional dimension. Geom. Funct. Anal. 19(2), 429–456 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Łaba I., Wang H.: Decoupling and near-optimal restriction estimates for Cantor sets. Int. Math. Res. Not. (2017, to appear)

    Google Scholar 

  21. Lyons, R., Peres, Y.: Probability on trees and networks. Cambridge Series in Statistical and Probabilistic Mathematics, vol. 42, xv+699 pp. Cambridge University Press, New York (2016)

    Google Scholar 

  22. Mandelbrot, B.B.: Renewal sets and random cutouts. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 22, 145–157 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mandelbrot, B.B.: Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier. J. Fluid. Mech. 62, 331–358 (1974)

    Article  MATH  Google Scholar 

  24. Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces. Cambridge Studies in Advanced Mathematics, vol. 44. Cambridge University Press, Cambridge (1995). Fractals and rectifiability

    Google Scholar 

  25. Mitsis, T.: A Stein-Tomas restriction theorem for general measures. Publ. Math. Debr. 60(1-2), 89–99 (2002)

    MathSciNet  MATH  Google Scholar 

  26. Mockenhaupt, G.: Salem sets and restriction properties of Fourier transforms. Geom. Funct. Anal. 10(6), 1579–1587 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  27. Peres, Y., Rams, M.: Projections of the natural measure for percolation fractals. Israel J. Math. 214(2), 539–552 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  28. Rams, M., Simon, K.: The dimension of projections of fractal percolations. J. Stat. Phys. 154(3), 633–655 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Rams, M., Simon, K.: Projections of fractal percolations. Ergodic Theory Dyn. Syst. 35(2), 530–545 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  30. Salem, R.: On singular monotonic functions whose spectrum has a given Hausdorff dimension. Ark. Mat. 1, 353–365 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  31. Shmerkin, P.: Salem sets with no arithmetic progressions. Int. Math. Res. Not. 2017(7), 1929–1941 (2017)

    MathSciNet  Google Scholar 

  32. Shmerkin, P., Suomala, V.: Sets which are not tube null and intersection properties of random measures. J. Lond. Math. Soc. (2) 91(2), 405–422 (2015)

    Google Scholar 

  33. Shmerkin, P., Suomala, V.: Spatially independent martingales, intersections, and applications. Mem. Am. Math. Soc. Available at http://arxiv.org/abs/1409.6707 (to appear, 2015)

  34. Shmerkin, P., Suomala, V.: Patterns in random fractals (2016). Preprint, available at https://arxiv.org/abs/1703.09553

  35. Simon, K., Vágó, L.: Projections of Mandelbrot percolation in higher dimensions. In: Fractals, Wavelets, and Their Applications. Springer Proceedings in Mathematics and Statistics, vol. 92, pp. 175–190. Springer, Cham (2014)

    Google Scholar 

  36. Tricot, C. Jr.: Two definitions of fractional dimension. Math. Proc. Camb. Philos. Soc. 91(1), 57–74 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  37. Wolff, T.H.: Lectures on Harmonic Analysis. University Lecture Series, vol. 29. American Mathematical Society, Providence (2003). With a foreword by Charles Fefferman and preface by Izabella Łaba, Edited by Łaba and Carol Shubin

    Google Scholar 

Download references

Acknowledgements

We thank Julien Barral for useful comments on an earlier version of the article. Pablo Shmerkin was partially supported by Projects PICT 2013-1393 and PICT 2014-1480 (ANPCyT). Ville Suomala acknowledges support from the Centre of Excellence in Analysis and Dynamics Research funded by the Academy of Finland.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Torcuato Di Tella University, Buenos Aires, Argentina

    Pablo Shmerkin

  2. CONICET, Buenos Aires, Argentina

    Pablo Shmerkin

  3. Department of Mathematical Sciences, University of Oulu, Oulu, Finland

    Ville Suomala

Authors
  1. Pablo Shmerkin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ville Suomala
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pablo Shmerkin .

Editor information

Editors and Affiliations

  1. Laboratoire Analyse, Géométrie et Applications, Université Paris 13, Villetaneuse, France

    Julien Barral

  2. Laboratoire d’Analyse et de Mathématiques Appliquées (UMR 8050), Université Paris-Est, UPEMLV, UPEC, CNRS, Créteil, France

    Stéphane Seuret

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Shmerkin, P., Suomala, V. (2017). A Class of Random Cantor Measures, with Applications. In: Barral, J., Seuret, S. (eds) Recent Developments in Fractals and Related Fields. FARF3 2015. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-57805-7_11

Download citation

Publish with us

Policies and ethics

Search

Navigation