Erdős's reverse Littlewood–Offord conjecture

Let $x_1,\dotsc,x_n$ be unit complex numbers, and let $\varepsilon_1,\dotsc,\varepsilon_n\in\{-1,+1\}$. Erdős's conjecture. The number of sums

$$\sum_{i=1}^n \varepsilon_i x_i$$

with

$$\left|\sum_{i=1}^n \varepsilon_i x_i\right|\leq 1$$

is greater than $c2^n/n$ for some absolute constant $c>0$. This conjecture is false as stated: taking an odd number of copies of $(1,0)$ and of $(0,1)$ makes every sum have norm at least $\sqrt{2}$, so the unit-radius claim cannot hold in general.