The Reverse Littlewood--Offord problem of Erdos

Let ₁, , ₍ be a sequence of independent Rademacher random variables. We prove that there is a constant c>0 such that for any unit vectors v₁, , vₙ R², ||₁ v₁++ₙ vₙ||₂ 2 cn. This resolves the only remaining conjecture from the seminal paper of Erdos on the Littlewood--Offord problem, and it is sharp both in the sense that the constant 2 cannot be reduced and that the magnitude n^-1 is best possible. We also prove polynomial bounds for the analogous problem in higher dimensions.