Erdős Problem #498
strong Littlewood-Offord problem

Let \(z_1,\ldots,z_n\in\mathbb{C}\) with \(1\leq \lvert z_i\rvert\) for \(1\leq i\leq n\). Let \(D\) be an arbitrary disc of radius \(1\). Is it true that the number of sums of the shape \[\sum_{i=1}^n\epsilon_iz_i \textrm{ for }\epsilon_i\in \{-1,1\}\] which lie in \(D\) is at most \(\binom{n}{\lfloor n/2\rfloor}\)? A strong form of the Littlewood-Offord problem. Erdős [Er45] proved this is true if \(z_i\in\mathbb{R}\), and for general \(z_i\in\mathbb{C}\) proved a weaker upper bound of \[\ll \frac{2^n}{\sqrt{n}}.\] This was solved in the affirmative by Kleitman [Kl65], who also later generalised this to arbitrary Hilbert spaces [Kl70]. See also [395].