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Small Ball Probability, Inverse Theorems, and Applications

  • Chapter
Erdős Centennial

Part of the book series: Bolyai Society Mathematical Studies ((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.

The first author is supported by research grant DMS-1256802.

The second author is supported by research grants DMS-0901216 and AFOSAR-FA-9550-12-1-0083.

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Authors and Affiliations

  1. Department of Mathematics, Yale University, 10 Hillhouse Ave., New Haven, CT, 06511, USA

    Hoi H. Nguyen & Van H. Vu

Authors
  1. Hoi H. Nguyen
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  2. Van H. Vu
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Editors and Affiliations

  1. Department of Computer Science, Eötvös Lóránd University, Pázmány P. sétány 1/c, Budapest, 1117, Hungary

    László Lovász

  2. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u. 13–15, Budapest, 1053, Hungary

    Imre Z. Ruzsa  & Vera T. Sós  & 

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Nguyen, H.H., Vu, V.H. (2013). Small Ball Probability, Inverse Theorems, and Applications. In: Lovász, L., Ruzsa, I.Z., Sós, V.T. (eds) Erdős Centennial. Bolyai Society Mathematical Studies, vol 25. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39286-3_16

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