PROVED
This has been solved in the affirmative.
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].
View the LaTeX source
Additional thanks to: Zach Hunter
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #395, https://www.erdosproblems.com/395, accessed 2026-01-16
