Advertisement

Geometric and Functional Analysis

, Volume 19, Issue 2, pp 429–456 | Cite as

Arithmetic Progressions in Sets of Fractional Dimension

  • Izabella Łaba
  • Malabika Pramanik
Article

Abstract

Let \({E \subset\mathbb{R}}\) be a closed set of Hausdorff dimension α. Weprove that if α is sufficiently close to 1, and if E supports a probability measure obeying appropriate dimensionality and Fourier decay conditions, then E contains non-trivial 3-term arithmetic progressions.

Keywords and phrases

Arithmetic progressions Salem sets Hausdorff dimension restriction estimates 

2000 Mathematics Subject Classification

28A78 42A32 42A38 42A45 11B25 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. B.
    Behrend F.A. (1946) On sets of integers which contain no three terms in arithmetical progression. Proc. Nat. Acad. Sci. USA 32: 331–332MathSciNetCrossRefzbMATHGoogle Scholar
  2. Bl1.
    Bluhm C. (1996) Random recursive construction of Salem sets. Ark. Mat. 34: 51–63MathSciNetCrossRefzbMATHGoogle Scholar
  3. Bl2.
    Bluhm C. (1998) On a theorem of Kaufman: Cantor-type construction of linear fractal Salem sets. Ark. Mat. 36: 307–316MathSciNetCrossRefzbMATHGoogle Scholar
  4. Bo1.
    Bourgain J. (1987) Construction of sets of positive measure not containing an affine image of a given infinite structure. Israel J. Math. 60: 333–344MathSciNetCrossRefzbMATHGoogle Scholar
  5. Bo2.
    Bourgain J. (1989) On Λ(p)-subsets of squares. Israel J. Math. 67: 291–311MathSciNetCrossRefzbMATHGoogle Scholar
  6. Bo3.
    Bourgain J. (1993) Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, I, GAFA. Geom. funct. anal. 3: 107–156MathSciNetCrossRefzbMATHGoogle Scholar
  7. DPRZ.
    Dembo A., Peres Y., Rosen J., Zeitouni O. (2001) Thick points for planar Brownian motion and the Erdős–Taylor conjecture on random walk. Acta. Math. 186: 239–270MathSciNetCrossRefzbMATHGoogle Scholar
  8. E.
    P. Erdős, My Scottish Book “problems”, in “The Scottish Book” (R.D. Mauldin, ed.), Birkhäuser, Boston (1981).Google Scholar
  9. F1.
    Falconer K. (1984) On a problem of Erdős on sequences and measurable sets. Proc. Amer. Math. Soc. 90: 77–78MathSciNetzbMATHGoogle Scholar
  10. F2.
    K. Falconer, The Geometry of Fractal sets, Cambridge Univ. Press 1985.Google Scholar
  11. G1.
    Green B. (2002) Arithmetic progressions in sumsets, GAFA. Geom. funct. anal. 12: 584–597MathSciNetCrossRefzbMATHGoogle Scholar
  12. G2.
    Green B. (2005) Roth’s theorem in the primes. Ann. Math. 161: 1609–1636CrossRefzbMATHGoogle Scholar
  13. GT1.
    Green B., Tao T. (2008) The primes contain arbitrarily long arithmetic progressions. Ann. Math. 167: 481–547MathSciNetCrossRefzbMATHGoogle Scholar
  14. GT2.
    Green B., Tao T. (2006) Restriction theory of the Selberg sieve, with applications. J. Théor. Nombres Bordeaux 18: 147–182MathSciNetzbMATHGoogle Scholar
  15. HL.
    Humke P.D., Laczkovich M. (1998) A visit to the Erdős problem. Proc. Amer. Math. Soc. 126: 819–822MathSciNetCrossRefzbMATHGoogle Scholar
  16. K.
    J.P. Kahane, Some Random Series of Functions, Cambridge Univ. Press, 1985.Google Scholar
  17. Ka.
    Kaufman L. (1981) On the theorem of Jarnik and Besicovitch. Acta Arith. 39: 265–267MathSciNetzbMATHGoogle Scholar
  18. Ke.
    T. Keleti, A 1-dimensional subset of the reals that intersects each of its translates in at most a single point, Real Anal. Exchange 24:2 (1998/99), 843–844.Google Scholar
  19. KoŁR.
    KohayakawaY., Łuczak T., Rödl V. (1996) Arithmetic progressions of length three in subsets of a random set. Acta Arith. 75: 133–163MathSciNetzbMATHGoogle Scholar
  20. Kol.
    Kolountzakis M. (1997) Infinite patterns that can be avoided by measure. Bull. London Math. Soc. 29(4): 415–424MathSciNetCrossRefzbMATHGoogle Scholar
  21. Kóm.
    Kómjáth P. (1983) Large sets not containing images of a given sequence. Canad. Math. Bull. 26: 41–43MathSciNetCrossRefzbMATHGoogle Scholar
  22. M.
    P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge Studies in Advanced Mathematics 44, Cambridge University Press, 1995.Google Scholar
  23. Mo.
    Mockenhaupt G. (2000) Salem sets and restriction properties of Fourier transforms. GAFA, Geom. funct. anal. 10: 1579–1587MathSciNetCrossRefzbMATHGoogle Scholar
  24. R.
    Roth K. (1953) On certain sets of integers. J. London Math. Soc. 28: 245–252MathSciNetCrossRefGoogle Scholar
  25. S.
    Salem R. (1950) On singular monotonic functions whose spectrum has a given Hausdorff dimension. Ark. Mat. 1: 353–365MathSciNetCrossRefGoogle Scholar
  26. SS.
    Salem R., Spencer D.C. (1942) On sets of integers which contain no three terms in arithmetical progression. Proc. Nat. Acad. Sci. USA 28: 561–563MathSciNetCrossRefzbMATHGoogle Scholar
  27. St1.
    E.M. Stein, Oscillatory integrals in Fourier analysis, in “Beijing Lectures in Harmonic Analysis (E.M. Stein, ed.), Ann. Math. Study 112, Princeton Univ. Press (1986), 307–355.Google Scholar
  28. St2.
    Stein E.M. (1993) Harmonic Analysis. Princeton Univ. Press, PrincetonzbMATHGoogle Scholar
  29. T1.
    T. Tao, Arithmetic progressions and the primes, Collect. Math. Extra Vol. (2006), 37–88.Google Scholar
  30. TV.
    T. Tao, V. Vu, Additive Combinatorics, Cambridge University Press, 2006.Google Scholar
  31. To1.
    Tomas P.A. (1975) A restriction theorem for the Fourier transform. Bull. Amer. Math. Soc. 81: 477–478MathSciNetCrossRefzbMATHGoogle Scholar
  32. To2.
    P.A. Tomas, Restriction theorems for the Fourier transform, in “Harmonic Analysis in Euclidean Spaces” (G. Weiss, S. Wainger, eds.), Proc. Symp. Pure Math. 35:I, Amer. Math. Soc. (1979), 111–114.Google Scholar
  33. W.
    T.Wolff, Lectures on Harmonic Analysis (I. Łaba, C. Shubin, eds.), Amer. Math. Soc., Providence, R.I. (2003).Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of British ColumbiaVancouverCanada

Personalised recommendations