Optimal Littlewood-Offord Inequalities in Groups

Consider the weighted sum [[summation].sup.n.sub.i=1] [a.sub.i][[epsilon].sub.i], where [([a.sub.i]).sup.n.sub.i=1] is a sequence of non-zero real numbers and [([[epsilon].sub.i]).sup.n.sub.i=1]--a sequence of independent Rademacher random variables, that is, P([[epsilon].sub.i[+ or -]1]) = -1. The classical Littlewood-Offord problem asks for the best possible upper bound for the probability P ([[summation].sup.n.sub.i=1] [a.sub.i][[epsilon].sub.i] = X. The first optimal result was obtained by Erdos in 1943 and since then his result was extended in many directions by Frankl, Katona, Kleitman, Griggs and other authors. In this paper we prove several Littlewood-Offord type inequalities in arbitrary groups. In groups having elements of finite order the worst case scenario is provided by the simple random walk on a cyclic subgroup. The inequalities we obtain are optimal if the underlying group contains an element of a certain order. It turns out that for torsion-free groups Erdos's bound still holds. Our results strengthen and generalize some very recent results by Tiep and Vu for certain matrix groups.