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

Let $\epsilon_{1},\ldots,\epsilon_{n}$ be a sequence of independent Rademacher random variables. We prove that there is a constant $c>0$ such that for any unit vectors $v_1,\ldots,v_n\in \mathbb{R}^2$, $$\Pr\left[||\epsilon_1 v_1+\ldots+\epsilon_n v_n||_2 \leq \sqrt{2}\right]\geq \frac{c}{n}.$$ This resolves the only remaining conjecture from the seminal paper of Erd\H{o}s on the Littlewood--Offord problem, and it is sharp both in the sense that the constant $\sqrt{2}$ cannot be reduced and that the magnitude $n^{-1}$ is best possible. We also prove polynomial bounds for the analogous problem in higher dimensions.