PROVED
This has been solved in the affirmative.
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].
View the LaTeX source
Additional thanks to: Stijn Cambie
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 #498, https://www.erdosproblems.com/498, accessed 2026-01-16
