Theorem 14 says that if $f$ is an unbiased linear threshold function $f(x) = \mathrm{sgn}(a_1 x_1 + \cdots + a_n x_n)$ in which all $a_i$’s are “small” then the noise stability $\mathbf{Stab}_\rho[f]$ is at least (roughly) $\frac{2}{\pi} \arcsin \rho$. Rephrasing in terms of noise sensitivity, this means $\mathbf{NS}_\delta[f]$ is at most (roughly) $\tfrac{2}{\pi} \sqrt{\delta} [...]

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Ohad Klein: 41a (45a in book): "let T be ...; prove something about f" ...Ryan O'Donnell: Good catch, thank you Xi.Ryan O'Donnell: Thank you! Sorry for the delay in replying.Ryan O'Donnell: Hi Ming. Here S stands for a fixed (non-random) subset of [...Xi Wu: typo: "our definition of $\mathbf{Inf}_i[f]$ from Chapter 2....Chengyu: Ex 2.c It should be "Suppose ... is an LTF with $\textbf{E}...Ming: I confuse the notation S in Fact 1.7. I wonder that the sym...