Majority Is Asymptotically the Most Stable Resilient Function

The result that `Majority is Stablest, proven with O'Donnell and Oleszkiewicz (2005), states that, asymptotically, among all Boolean functions with sufficiently low influences and mean $0$, a simple majority function is most stable as the number of variables goes to infinity. It is natural to ask if the condition of low influences can be relaxed to the condition that the function has vanishing Fourier coefficients. Here we answer this question affirmatively. In fact, we prove a stronger statement showing that any Boolean function $f : \{0,1\}^n \to [0,1]$ with $E[f] = 1/2$ which is $\epsilon > 0$ more stable than the asymptotic stability of a balanced majority has correlation at least $\alpha(\epsilon) > 0$ with a Boolean function $g$ which depends on a bounded number, $r(\epsilon)$, of variables