Small ball probability, Inverse theorems, and applications

2012/12/31 by Hoi H. Nguyen, Nguyen, Hoi H., Van H. Vu +1 · 4 citations
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics #math.CO #math.PR

paper · pdf · doi:10.48550/arxiv.1301.0019

47 pages

arxiv created 2012/12/31 · openalex publication_date 2012/12/31 · arxiv updated 2013/01/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let ξ be a real random variable with mean zero and variance one and A=a₁,...,a_n be a multi-set in \R^d. The random sum S_A := a₁ ξ₁ + ... + a_n ξ_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 Erdos 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.

Cited by

• Uniform spectral gaps, non-abelian Littlewood-Offord and anti-concentration for random walks
• Geometric Littlewood-Offord problems via lattice point counting
• Algebraic aspects of the polynomial Littlewood-Offord problem
• Limit law for root separation in random polynomials

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• The Littlewood-Offord problem in high dimensions and a conjecture of Frankl and Füredi
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• Inverse Littlewood-Offord theorems and the condition number of random discrete matrices