Israel Journal of Mathematics

, Volume 179, Issue 1, pp 253–278 | Cite as

The structure of popular difference sets

Article

Abstract

We show that the set of popular differences of a large subset of ℤN does not always contain the complete difference set of another large set. For this purpose we construct a so-called niveau set, which was first introduced by Ruzsa in [Ruz87] and later used in [Ruz91] to show that there exists a large subset of ℤN whose sumset does not contain any long arithmetic progressions. In this paper we make substantial use of measure concentration results on the multi-dimensional torus and Esseen’s Inequality.

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References

  1. [AK94]
    W. Albers and W. C. M. Kallenberg, A simple approximation to the bivariate normal distribution with large correlation coefficient, Journal of Multivariate Analysis 49 (1994), 87–96.MATHCrossRefMathSciNetGoogle Scholar
  2. [AS00]
    N. Alon and J. Spencer, The Probabilistic Method, Wiley-Interscience [John Wiley & Sons], New York, 2000.MATHCrossRefGoogle Scholar
  3. [Ber41]
    A. C. Berry, The accuracy of the Gaussian approximation to the sum of independent variates, Transactions of the American Mathematical Society 49 (1941), 122–136.MATHMathSciNetGoogle Scholar
  4. [Ber45]
    H. Bergstrom, On the central limit theorem in the casek, k > 1, Skand. Aktuarietidskr. 2 (1945), 106–127.MathSciNetGoogle Scholar
  5. [DT97]
    M. Drmota and R. F. Tichy, Sequences, Discrepancies and Applications, Springer-Verlag, Berlin, 1997.MATHGoogle Scholar
  6. [Ess45]
    C-G. Esseen, Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian law, Acta Mathematica 77 (1945), 1–125.MATHCrossRefMathSciNetGoogle Scholar
  7. [Gre03]
    B. J. Green, Some constructions in the inverse spectral theory of cyclic groups, Combinatorics, Probability and Computing 12 (2003), 127–138.MATHCrossRefMathSciNetGoogle Scholar
  8. [Gre05]
    B. J. Green, Finite field models in additive combinatorics, in Surveys in combinatorics 2005, London Math. Soc. Lecture Note Ser., vol. 327, Cambridge University Press, Cambridge, 2005, pp. 1–27.CrossRefGoogle Scholar
  9. [GR05]
    B. J. Green and I. Ruzsa, Sum-free sets in abelian groups, Israel Journal of Mathematics 147 (2005), 157–189.MATHCrossRefMathSciNetGoogle Scholar
  10. [Har66]
    L. H. Harper, Optimal numberings and isoperimetric problems on graphs, Journal of Combinatorial Theory 1 (1966), 385–393.MATHCrossRefMathSciNetGoogle Scholar
  11. [JŁR00]
    S. Janson, T. Łuczak and A. Rucinski, Random Graphs, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley-Interscience [John Wiley & Sons], New York, 2000.MATHGoogle Scholar
  12. [Kle66]
    D. Kleitman, On a combinatorial conjecture by Erdös, Journal of Combinatorial Theory 1 (1966), 209–214.MATHCrossRefMathSciNetGoogle Scholar
  13. [Led01]
    M. Ledoux, The concentration of measure phenomenon, AMS Mathematical Surveys and Monographs, vol. 89, American Mathematical Society, Providence, RI, 2001.MATHGoogle Scholar
  14. [LŁS01]
    V. Lev, T. Łuczak and T. Schoen, Sum-free sets in abelian groups, Israel Journal of Mathematics 125 (2001), 347–367.MATHCrossRefMathSciNetGoogle Scholar
  15. [McD89]
    C. McDiarmid, On the method of bounded differences, in Surveys in combinatorics, 1989 (Norwich, 1989), London Math. Soc. Lecture Note Ser., vol. 141, Cambridge University Press, Cambridge, 1989, pp. 148–188.Google Scholar
  16. [NP73]
    H. Niederreiter and W. Philipp, Berry-Esseen bounds and a theorem of Erdös and Turán on uniform distribution mod 1, Duke Mathematical Journal 40 (1973), 633–649.MATHCrossRefMathSciNetGoogle Scholar
  17. [Ruz87]
    I. Z. Ruzsa, Essential components, Proceedings of the London Mathematical Society. Third Series 54 (1987), 38–56.MATHCrossRefMathSciNetGoogle Scholar
  18. [Ruz91]
    I. Z. Ruzsa, Arithmetic progressions in sumsets, Acta Arithmetica 2 (1991), 191–202.MathSciNetGoogle Scholar
  19. [Sad66]
    S. M. Sadikova, Two-dimensional analogues of an inequality of Esseen with applications to the Central Limit Theorem, Theory of Probability and Its Applications 11 (1966), 325–335.CrossRefGoogle Scholar
  20. [San08]
    T. Sanders, Popular difference sets, Available at http://arxiv.org/abs/0807.5106, 2008.
  21. [Shi84]
    A. N. Shiryayev, Probability, Graduate Texts in Mathematics, vol. 95, Springer-Verlag, New York, 1984.Google Scholar

Copyright information

© Hebrew University Magnes Press 2010

Authors and Affiliations

  1. 1.Department of Pure Mathematics and Mathematical StatisticsCambridgeUK
  2. 2.Institute for Advanced StudySchool of MathematicsPrincetonUSA

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