Optimal Littlewood-Offord inequalities in groups
arXiv:1707.01085 · doi:10.1007/s00493-018-3845-7
Abstract
We prove several Littlewood-Offord type inequalities for arbitrary groups. In groups having elements of finite order the worst case scenario is provided by the simple random walk on a certain cyclic subgroup. The inequalities we obtain are optimal if the underlying group contains an element of certain order. It turns out that for torsion-free groups Erdős's bound still holds. Our results strengthen and generalize some very recent results by Tiep and Vu.