The Reverse Littlewood--Offord problem of Erd{o}s

Let ε₁,…,εₙ 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, [||ε₁ v₁+…+εₙ vₙ||₂ ≤ √2]≥ c/n. This resolves the only remaining conjecture from the seminal paper of Erd{o}s 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.