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Small Ball Probability, Inverse Theorems, and Applications

  • Hoi H. Nguyen
  • Van H. Vu
Part of the Bolyai Society Mathematical Studies book series (BSMS, volume 25)

Abstract

Let ξ be a real random variable with mean zero and variance one and A ={a 1; …; a n } be a multi-set in R d . The random sum
$$S_A : = a_1 \xi _1 + \cdots + a_n \xi _n$$
where ξ i are iid copies of ξ is of fundamental importance in probability and its applications.

We discuss the small ball problem, the aim of which is to estimate the maximum probability that S A belongs to a ball with given small radius, following the discovery made by Littlewood-Offord and Erdős almost 70 years ago. We will mainly focus on recent developments that characterize the structure of those sets A where the small ball probability is relatively large. Applications of these results include full solutions or significant progresses of many open problems in different areas.

Keywords

Random Matrice Span Versus Algebraic Degree Inverse Theorem Random Polynomial 
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.

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Copyright information

© János Bolyai Mathematical Society and Springer-Verlag 2013

Authors and Affiliations

  • Hoi H. Nguyen
    • 1
  • Van H. Vu
    • 1
  1. 1.Department of MathematicsYale UniversityNew HavenUSA

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