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On a lemma of Littlewood and Offord

  • Martin Aigner
  • Günter M. Ziegler

Abstract

In their work on the distribution of roots of algebraic equations, Littlewood and Offord proved in 1943 the following result:
Let a 1, a 2,..., a n be complex numbers with |a 2| ≥ 1 for all i, and consider the 2 n linear combinations \( \sum\nolimits_{i = 1}^n {\varepsilon _i a_i \;with\;\varepsilon _i \in \left\{ {1, - 1} \right\}} \) with ε i ∈ {1, −1}. Then the number of sums \( \sum\nolimits_{i = 1}^n {\varepsilon _i a_i } \) which lie in the interior of any circle of radius 1 is not greater than
$$ c\frac{{2^n }} {{\sqrt n }}\log n\;for\;some\;constant\;c > 0. $$

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References

  1. [1]
    P. Erdős: On a lemma of Littlewood and Offord, Bulletin Amer. Math. S 51 (1945), 898–902.CrossRefGoogle Scholar
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    G. Kätona: On a conjecture of Er do s and a stronger form of Sperne theorem, Studia Sci. Math. Hungar. 1 (1966), 59–63.MathSciNetGoogle Scholar
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    D. Kleitman: On a lemma of Littlew ood and Offord on the distribution certain sums, Math. Zeitschrift 90 (1965), 251–259.MathSciNetzbMATHCrossRefGoogle Scholar
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    D. Kleitman: On a lemma of Littlew ood and Offord on the distributions linear combinations of vectors, Advances Math. 5 (1970), 155–157.MathSciNetzbMATHCrossRefGoogle Scholar
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    J. E. Littlewood & A. C. Offord: On the number of real roots o, random algebraic equation 111, Mat. Ussr Sb. 12 (1943), 277–285.MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Martin Aigner
    • 1
  • Günter M. Ziegler
    • 2
  1. 1.Institut für Mathematik II (WE2)Freie Universität BerlinBerlinGermany
  2. 2.Institut für Mathematik, MA 6-2Technische Universität BerlinBerlinGermany

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