Stability of the Courtade-Kumar inequality

Abstract

We prove dimension-independent stability for the Courtade-Kumar inequality: a Boolean function $f:\{-1,1{\}}^n\to \{-1,1\}$ whose information is close to the dictator value is close in probability to a signed dictator. The correlation dependence is sharp in order near zero and, for increasing functions, also at the noiseless endpoint.

18 pages