Let ξ be a real random variable with mean zero and variance one and
A=a1,...,an be a multi-set in Rd. The random sum
SA:=a1ξ1+...+anξn where ξi 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 SA 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.Comment: 47 page