On some extensions of the FKN theorem
Revised: September 19, 2015
Published: December 29, 2015
$\newcommand{\Var}{{\mathrm {Var}}}$
Let $S=a_{1}r_{1}+a_{2}r_{2}+\ldots+a_{n}r_{n}$ be a weighted Rademacher sum. Friedgut, Kalai, and Naor have shown that if $\Var(|S|)$ is much smaller than $\Var(S)$, then the sum is largely determined by one of the summands. We provide a simple and elementary proof of this result, strengthen it, and extend it in various ways to a more general setting.