A new approach to an old problem of Erdős and Moser

Let ηi, i = 1, . . . , n be iid Bernoulli random variables, taking values ±1 with probability 1 2 . Given a multiset V of n elements v1, . . . , vn of an additive group G, we define the concentration probability of V as ρ(V ) := sup v∈G P(η1v1 + . . . ηnvn = v). An old result of Erdős and Moser asserts that if vi are distinct real numbers then ρ(V ) is O(n− 3 2 logn). This bound was then refined by Sárközy and Szemerédi 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.