Let ξ be a real random variable with mean zero and variance one and
} be a multi-set in
. The random sum
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$$S_A : = a_1 \xi _1 + \cdots + a_n \xi _n$$
are iid copies of ξ is of fundamental importance in probability and its applications.
We discuss the
problem, the aim of which is to estimate the maximum probability that
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
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.