Erdős Problem #395
reverse Littlewood-Offord problem

If \(z_1,\ldots,z_n\in \mathbb{C}\) with \(\lvert z_i\rvert=1\) then is it true that the probability that \[\lvert \epsilon_1z_1+\cdots+\epsilon_nz_n\rvert \leq \sqrt{2},\] where \(\epsilon_i\in \{-1,1\}\) uniformly at random, is \(\gg 1/n\)? A reverse Littlewood-Offord problem. Erdős originally asked this with \(\sqrt{2}\) replaced by \(1\), but Carnielli and Carolino [CaCa11] observed that this is false, choosing \(z_1=1\) and \(z_k=i\) for \(2\leq k\leq n\), where \(n\) is even, since then the sum is at least \(\sqrt{2}\) always. Solved in the affirmative by He, Juškevičius, Narayanan, and Spiro [HJNS24]. The bound of \(1/n\) is the best possible, as shown by taking \(z_k=1\) for \(1\leq k\leq n/2\) and \(z_k=i\) otherwise. See also [498](https://www.erdosproblems.com/498).