A new approach to an old problem of Erdős and Moser

Let η i , i = 1 , … , n be independent identically distributed Bernoulli random variables, taking values ±1 with probability 1 2 . Given a multiset V of n elements v 1 , … , v n of an additive group G, we define ρ ( V ) as ρ ( V ) : = sup v ∈ G P ( η 1 v 1 + ⋯ + η n v n = v ) . An old result of Erdős and Moser asserts that if G = R and the v i are distinct then ρ ( V ) is O ( n − 3 2 log n ) . 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 is due to Stanley who used tools from algebraic geometry to give a complete description for sets having optimal ρ ( V ) ; the result has become classic in algebraic combinatorics. s paper, we will prove that the optimal sets from Stanleyʼs work are stable. More importantly, our result gives an almost complete description for sets having large concentration probability.