A new approach to an old problem of Erdos and Moser

Let $\eta_i, i=1,..., n$ be iid Bernoulli random variables, taking values $\pm 1$ with probability 1/2. Given a multiset $V$ of $n$ elements $v_1, ..., v_n$ of an additive group $G$, we define the \emph{concentration probability} of $V$ as $$\rho(V) := \sup_{v\in G} P(\eta_1 v_1 + ... \eta_n v_n =v). $$ An old result of Erdos and Moser asserts that if $v_i $ are distinct real numbers then $\rho(V)$ is $O(n^{-3/2}\log n)$. This bound was then refined by Sarkozy and Szemeredi to $O(n^{-3/2})$, which is sharp up to a constant factor. The ultimate result dues to Stanley who used tools from algebraic geometry to give a complete description for sets having optimal concentration probability; the result now becomes classic in algebraic combinatorics. In this paper, we will prove that the optimal sets from Stanley's work are stable. More importantly, our result gives an almost complete description for sets having large concentration probability.