Having derived strong results about the Fourier spectrum of small DNFs and CNFs, we will now extend to the case of constant-depth circuits. We begin by describing how Håstad applied his Switching Lemma to constant-depth circuits. We then describe some Fourier-theoretic consequences coming from a very early (1989) work in analysis of boolean functions [...]

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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...