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The Probabilistic Method
[Benjamini et al. (Inst Hautes Études Sci Publ Math 90:5–43, 2001) showed that weighted majority functions of n independent unbiased bits are uniformly stable under noise: when each bit is flipped with probability 𝜖, the probability p𝜖 that the weighted majority changes is at most C𝜖1∕4. They asked what is the best possible exponent that could replace 1∕4. We prove that the answer is 1∕2. The upper bound obtained for p𝜖 is within a factor of π∕2+o(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\sqrt {\pi /2}+o(1)$$ \end{document} from the known lower bound when 𝜖 → 0 and n𝜖 →∞.]
Published: Nov 4, 2020
Keywords: Noise sensitivity; Boolean functions; Weighted majority; 60C05
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