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Arithmetic Progressions in Sets of Fractional Dimension

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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.

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References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Bluhm C. (1996) Random recursive construction of Salem sets. Ark. Mat. 34: 51–63

    Article  MathSciNet  MATH  Google Scholar 

  3. Bluhm C. (1998) On a theorem of Kaufman: Cantor-type construction of linear fractal Salem sets. Ark. Mat. 36: 307–316

    Article  MathSciNet  MATH  Google Scholar 

  4. Bourgain J. (1987) Construction of sets of positive measure not containing an affine image of a given infinite structure. Israel J. Math. 60: 333–344

    Article  MathSciNet  MATH  Google Scholar 

  5. Bourgain J. (1989) On Λ(p)-subsets of squares. Israel J. Math. 67: 291–311

    Article  MathSciNet  MATH  Google Scholar 

  6. Bourgain J. (1993) Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, I, GAFA. Geom. funct. anal. 3: 107–156

    Article  MathSciNet  MATH  Google Scholar 

  7. 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: 239–270

    Article  MathSciNet  MATH  Google Scholar 

  8. P. Erdős, My Scottish Book “problems”, in “The Scottish Book” (R.D. Mauldin, ed.), Birkhäuser, Boston (1981).

  9. Falconer K. (1984) On a problem of Erdős on sequences and measurable sets. Proc. Amer. Math. Soc. 90: 77–78

    MathSciNet  MATH  Google Scholar 

  10. K. Falconer, The Geometry of Fractal sets, Cambridge Univ. Press 1985.

  11. Green B. (2002) Arithmetic progressions in sumsets, GAFA. Geom. funct. anal. 12: 584–597

    Article  MathSciNet  MATH  Google Scholar 

  12. Green B. (2005) Roth’s theorem in the primes. Ann. Math. 161: 1609–1636

    Article  MATH  Google Scholar 

  13. Green B., Tao T. (2008) The primes contain arbitrarily long arithmetic progressions. Ann. Math. 167: 481–547

    Article  MathSciNet  MATH  Google Scholar 

  14. Green B., Tao T. (2006) Restriction theory of the Selberg sieve, with applications. J. Théor. Nombres Bordeaux 18: 147–182

    MathSciNet  MATH  Google Scholar 

  15. Humke P.D., Laczkovich M. (1998) A visit to the Erdős problem. Proc. Amer. Math. Soc. 126: 819–822

    Article  MathSciNet  MATH  Google Scholar 

  16. J.P. Kahane, Some Random Series of Functions, Cambridge Univ. Press, 1985.

  17. Kaufman L. (1981) On the theorem of Jarnik and Besicovitch. Acta Arith. 39: 265–267

    MathSciNet  MATH  Google Scholar 

  18. 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.

  19. KohayakawaY., Łuczak T., Rödl V. (1996) Arithmetic progressions of length three in subsets of a random set. Acta Arith. 75: 133–163

    MathSciNet  MATH  Google Scholar 

  20. Kolountzakis M. (1997) Infinite patterns that can be avoided by measure. Bull. London Math. Soc. 29(4): 415–424

    Article  MathSciNet  MATH  Google Scholar 

  21. Kómjáth P. (1983) Large sets not containing images of a given sequence. Canad. Math. Bull. 26: 41–43

    Article  MathSciNet  MATH  Google Scholar 

  22. P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge Studies in Advanced Mathematics 44, Cambridge University Press, 1995.

  23. Mockenhaupt G. (2000) Salem sets and restriction properties of Fourier transforms. GAFA, Geom. funct. anal. 10: 1579–1587

    Article  MathSciNet  MATH  Google Scholar 

  24. Roth K. (1953) On certain sets of integers. J. London Math. Soc. 28: 245–252

    Article  MathSciNet  Google Scholar 

  25. Salem R. (1950) On singular monotonic functions whose spectrum has a given Hausdorff dimension. Ark. Mat. 1: 353–365

    Article  MathSciNet  Google Scholar 

  26. Salem R., Spencer D.C. (1942) On sets of integers which contain no three terms in arithmetical progression. Proc. Nat. Acad. Sci. USA 28: 561–563

    Article  MathSciNet  MATH  Google Scholar 

  27. 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.

  28. Stein E.M. (1993) Harmonic Analysis. Princeton Univ. Press, Princeton

    MATH  Google Scholar 

  29. T. Tao, Arithmetic progressions and the primes, Collect. Math. Extra Vol. (2006), 37–88.

  30. T. Tao, V. Vu, Additive Combinatorics, Cambridge University Press, 2006.

  31. Tomas P.A. (1975) A restriction theorem for the Fourier transform. Bull. Amer. Math. Soc. 81: 477–478

    Article  MathSciNet  MATH  Google Scholar 

  32. 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.

  33. T.Wolff, Lectures on Harmonic Analysis (I. Łaba, C. Shubin, eds.), Amer. Math. Soc., Providence, R.I. (2003).

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  1. Department of Mathematics, University of British Columbia, Vancouver, B.C., V6T 1Z2, Canada

    Izabella Łaba & Malabika Pramanik

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  1. Izabella Łaba
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  2. Malabika Pramanik
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Correspondence to Izabella Łaba.

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Łaba, I., Pramanik, M. Arithmetic Progressions in Sets of Fractional Dimension. Geom. Funct. Anal. 19, 429–456 (2009). https://doi.org/10.1007/s00039-009-0003-9

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