Small Ball Probability, Inverse Theorems, and Applications

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

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$$S_A : = a_1 \xi _1 + \cdots + a_n \xi _n$$

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.