Advertisement

Exponential Sums and Distinct Points on Arcs

  • Øystein J. RødsethEmail author
Chapter

Summary

Suppose that some harmonic analysis arguments have been invoked to show that the indicator function of a set of residue classes modulo some integer has a large Fourier coefficient. To get information about the structure of the set of residue classes, we then need a certain type of complementary result. A solution to this problem was given by Gregory Freiman in 1961, when he proved a lemma which relates the value of an exponential sum with the distribution of summands in semi-circles of the unit circle in the complex plane. Since then, Freiman’s Lemma has been extended by several authors. Rather than residue classes, one has considered the situation for finitely many arbitrary points on the unit circle. So far, Lev is the only author who has taken into consideration that the summands may be bounded away from each other, as is the case with distinct residue classes. In this paper, we extend Lev’s result by lifting a recent result of ours to the case of the points being bounded away from each other.

Keywords

Arcs Distribution Exponential sums Unit circle 

References

  1. 1.
    B. Green, Review MR2192089(2007a:11029) in MathSciNetGoogle Scholar
  2. 2.
    G. A. Freiman, Inverse problems of additive number theory. On the addition of sets of residues with respect to a prime modulus (Russian). Dokl. Akad. Nauk SSSR 141, 571–573 (1961)Google Scholar
  3. 3.
    G. A. Freiman, Inverse problems of additive number theory. On the addition of sets of residues with respect to a prime modulus. Sov. Math.-Dokl. 2, 1520–1522 (1961)Google Scholar
  4. 4.
    G. A. Freiman, Foundations of a Structural Theory of Set Addition. Translations of Mathematical Monographs, Vol. 37, American Mathematical Society, Providence, R. I., 1973Google Scholar
  5. 5.
    V. F. Lev, Distribution of points on arcs. Integers 5 (2),#A11, 6 pp. (electronic) (2005)Google Scholar
  6. 6.
    V. F. Lev, More on points and arcs. Integers 7 (2), #A24, 3 pp. (electronic) (2007)Google Scholar
  7. 7.
    V. F. Lev, Large sum-free sets in \(\mathbb{Z}/p\mathbb{Z}\). Israel J. Math 154, 221–234 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    W. Moran, A. D. Pollington, On a result of Freiman. Preprint 1992Google Scholar
  9. 9.
    M. B. Nathanson, Additive Number Theory: Inverse Problems and the Geometry of Sumsets. Graduate Texts in Mathematics, Vol. 165, Springer, New York, 1996Google Scholar
  10. 10.
    L. P. Postnikova, Fluctuations in the distribution of fractional parts. Dokl. Akad. Nauk SSSR, 161, 1282–1284 (1965)MathSciNetGoogle Scholar
  11. 11.
    Ø. J. Rødseth, Distribution of points on the circle. J. Number Theory 127, (1), 127–135 (2007)Google Scholar
  12. 12.
    Ø. J. Rødseth, Addendum to “Distribution of points on the circle”. J. Number Theory 128, (6), 1889–1892 (2008)Google Scholar
  13. 13.
    T. Tao, V. Vu, Additive Combinatorics. Cambridge Studies in Advanced Mathematics, Vol. 105, Cambridge University Press, 2006Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of BergenBergenNorway

Personalised recommendations