Advertisement

Ukrainian Mathematical Journal

, Volume 62, Issue 3, pp 380–419 | Cite as

On one result of J. Bourgain

  • S. V. Konyagin
  • I. D. Shkredov
Article
  • 66 Downloads

In a linear space of dimension n over the field \( {\mathbb{F}_2} \), we construct a set A of given density such that the Fourier transform of A is large on a large set, and the intersection of A with any subspace of small dimension is small. The results obtained show, in a certain sense, the sharpness of one theorem of J. Bourgain.

Keywords

Abelian Group Natural Number Nonnegative Integer Chebyshev Polynomial Linear Span 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    F. A. Behrend, “On sets of integers which contain no three terms in arithmetic progression,” Proc. Nat. Acad. Sci., 23, 331–332 (1946).CrossRefMathSciNetGoogle Scholar
  2. 2.
    P. Erdös and P. Turán, “On some sequences of integers,” J. London Math. Soc., 11, 261–264 (1936).zbMATHCrossRefGoogle Scholar
  3. 3.
    P. Frankl, G. Graham, and V. Rödl, “On sets of abelian groups with no 3-term arithmetic progressions,” J. Combin. Theory, 45, Ser. A, 157–161 (1987).zbMATHCrossRefGoogle Scholar
  4. 4.
    H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton, New York (1981).zbMATHGoogle Scholar
  5. 5.
    H. Furstenberg, “Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions,” J. Anal. Math., 31, 204–256 (1977).zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    H. Furstenberg and Y. Katznelson, “An ergodic Szemerédi theorem for commuting transformations,” J. Anal. Math., 34, 275–291 (1978).zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    H. Furstenberg, D. Ornstein, and Y. Katznelson, “The ergodic theoretical proof of Szemerédi’s theorem,” Bull. Amer. Math. Soc., 7, No. 3, 527–552 (1982).CrossRefMathSciNetGoogle Scholar
  8. 8.
    W. T. Gowers, “Rough structure and classification,” in: Geom. Funct. Anal., Special Volume GAFA2000 “Visions in Mathematics”, Tel Aviv, Part I, 79–117 (1999).Google Scholar
  9. 9.
    W. T. Gowers, “A new proof of Szemerédi’s theorem for arithmetic progressions of length four,” Geom. Funct. Anal., 8, 529–551 (1998).zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    W. T. Gowers, “A new proof of Szemerédi’s theorem,” Geom. Funct. Anal., 11, 465–588 (2001).zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    V. F. Lev, “Progressions-free sets in finite abelian groups,” J. Number Theory, 104, No. 1, 162–169 (2004).zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    R. Meshulam, “On subsets of finite abelian groups with no 3-term arithmetic progressions,” J. Combin. Theory, 71, Ser. A, 168–172 (1995).zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    M. B. Nathanson, Additive Number Theory. Inverse Problems and the Geometry of Sumsets, Springer, New York (1996).Google Scholar
  14. 14.
    R. A. Rankin, “Sets of integers containing not more than a given number of terms in arithmetic progression,” Proc. Roy. Soc. Edinburgh A, 65, No. 4, 332–344 (1961).MathSciNetGoogle Scholar
  15. 15.
    K. F. Roth, “On certain sets of integers,” J. London Math. Soc., 28, 245–252 (1953).CrossRefMathSciNetGoogle Scholar
  16. 16.
    M.-C. Chang, “A polynomial bound in Freiman’s theorem,” Duke Math. J., 113, No. 3, 399–419 (2002).zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    S. Bernstein, “Sur une modification de l’inéqualité de Tchebichef,” Ann. Sci. Inst. Sav. Ukr., Sect. Math. I (1924).Google Scholar
  18. 18.
    J. Spencer, “Six standard deviations suffice,” Trans. Amer. Math. Soc., 289, 679–706 (1985).zbMATHMathSciNetGoogle Scholar
  19. 19.
    B. Green, “Arithmetic progressions in sumsets,” Geom. Funct. Anal., 12, No. 3, 584–597 (2002).zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    B. Green, “A Szemerédi-type regularity lemma in abelian groups,” Geom. Funct. Anal., 15, No. 2, 340–376 (2005).zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    B. Green, “Some constructions in the inverse spectral theory of cyclic groups,” Combin. Probab. Comput., 12, No. 2, 127–138 (2003).zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    B. Green, “Spectral structure of sets of integers,” in: Fourier Analysis and Convexity (Applied and Numerical Harmonic Analysis) (Survey Article, Milan 2001), Birkhäuser, Boston (2004), pp. 83–96.Google Scholar
  23. 23.
    B. Green, “Structure theory of set addition,” in: ICMS Instructional Conference in Combinatorial Aspects of Mathematical Analysis (Edinburgh, March 25–April 5, 2002) (2002), pp. 1–27.Google Scholar
  24. 24.
    B. Green, “Finite field model in additive combinatorics,” Surveys in Combinatorics, London Mathematical Society Lecture Notes, 329, 1–29 (2005).Google Scholar
  25. 25.
    B. Green and T. Tao, “An inverse theorem for the Gowers U3-norm, with applications,” Proceedings of the Edinburgh Mathematical Society, Ser. 2, 51, No. 1, 73–153 (2008).zbMATHMathSciNetGoogle Scholar
  26. 26.
    B. Green and T. Tao, “New bounds for Szemerédi’s theorem, II: A new bound for r 4(N),” in: Analytic Number Theory, Cambridge University, Cambridge (2009), pp. 180–204.Google Scholar
  27. 27.
    W. Rudin, Fourier Analysis on Groups, Wiley, New York (1990)zbMATHGoogle Scholar
  28. 28.
    W. Rudin, “Trigonometric series with gaps,” J. Math. Mech., 9, 203–227 (1960).zbMATHMathSciNetGoogle Scholar
  29. 29.
    I. Ruzsa, “Arithmetic progressions in sumsets,” Acta Arithm., 60, No. 2, 191–202 (1991).zbMATHMathSciNetGoogle Scholar
  30. 30.
    T. Sanders, “Appendix to ‘Roth’s theorem on progressions revisited’ by J. Bourgain,” J. Anal. Math., 104, No. 1, 193–206 (2008).zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    E. Szemerédi, “On sets of integers containing no k elements in arithmetic progression,” Acta Arithm., 27, 299–345 (1975).Google Scholar
  32. 32.
    T. Tao and V. Vu, Additive Combinatorics, Cambridge University, Cambridge (2006).zbMATHCrossRefGoogle Scholar
  33. 33.
    T. Tao, Lecture Notes 5 for Math 254A. UCLA 2003; available at http://math.ucla.edu/~tao/254a.1.03w/notes5.dvi.
  34. 34.
    J. Bourgain, “On triples in arithmetic progression,” Geom. Funct. Anal., 9, 968–984 (1999).zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    J. Bourgain, Roth’s Theorem on Progressions Revisited, Preprint (2007).Google Scholar
  36. 36.
    G. Szegö, Orthogonal Polynomials, American Mathematical Society, New York (1939).Google Scholar
  37. 37.
    T. J. Rivlin, Chebyshev Polynomials: From Approximation Theory to Algebra and Number Theory, Wiley, New York (1990).zbMATHGoogle Scholar
  38. 38.
    P. Turan, “Eine Extremalaufgabe aus der Graphentheorie,” Mat. Fiz. Lapok., 48, 436–452 (1941).zbMATHMathSciNetGoogle Scholar
  39. 39.
    B. Bollobás, “Ket fuggelten kort nem tartalmazo grafokrol,” Mat. Lapok, 14, 311–321 (1963).zbMATHMathSciNetGoogle Scholar
  40. 40.
    B. Bollobás, Extremal Graph Theory, Academic Press, New York (1978).zbMATHGoogle Scholar
  41. 41.
    R. D. Dutton and R. C. Brigham, “Edges in graphs with large girth,” Graphs Combinatorics, 7, 315–321 (1991).zbMATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    I. D. Shkredov, “On sets of large trigonometric sums,” Dokl. Akad. Nauk SSSR, 411, No. 4, 455–459 (2006).MathSciNetGoogle Scholar
  43. 43.
    I. D. Shkredov, “On sets of large trigonometric sums,” Izv. Ros. Akad. Nauk, Ser. Mat., 72, No. 1, 161–182 (2008).MathSciNetGoogle Scholar
  44. 44.
    I. D. Shkredov, “Some examples of sets of large trigonometric sums,” Mat. Sb., 198, No. 12, 105–140 (2007).MathSciNetGoogle Scholar
  45. 45.
    I. D. Shkredov, “On sumsets of dissociated sets,” Online J. Anal. Combinatorics, 4, 1–27 (2009).MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2010

Authors and Affiliations

  • S. V. Konyagin
    • 1
  • I. D. Shkredov
    • 2
  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia
  2. 2.Lomonosov Moscow State UniversityMoscowRussia

Personalised recommendations