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## §10.3: Applications of general hypercontractivity

In this section we will collect some applications of the General Hypercontractivity Theorem, including generalizations of the facts from Section 9.5.

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## §10.1: The Hypercontractivity Theorem for uniform $\pm 1$ bits

In this section we’ll prove the full Hypercontractivity Theorem for uniform $\pm 1$ bits stated at the beginning of Chapter 9:

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## §9.6: Highlight: The Kahn–Kalai–Linial Theorem

Recalling the social choice setting of Chapter 2.5, consider a $2$-candidate, $n$-voter election using a monotone voting rule $f : \{-1,1\}^n \to \{-1,1\}$. We assume the impartial culture assumption (that the votes are independent and uniformly random), but with a twist: one of the candidates, say $b \in \{-1,1\}$, is able to secretly bribe $k$ [...]

## §9.5: Applications of hypercontractivity

With the $(2,q)$- and $(p,2)$-Hypercontractivity Theorems in hand, let’s revisit some applications we saw in Sections 1 and 2.

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## §9.2: Small subsets of the hypercube are noise-sensitive

An immediate consequence of the Bonami Lemma is that for any $f : \{-1,1\}^n \to {\mathbb R}$ and $k \in {\mathbb N}$, $$\label{eqn:2-4-hypercon-k} \|\mathrm{T}_{1/\sqrt{3}} f^{=k}\|_4 = \tfrac{1}{\sqrt{3}^k} \|f^{=k}\|_4 \leq \|f^{=k}\|_2.$$

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## §5.4: Degree-1 weight

In this section we prove two theorems about the degree-$1$ Fourier weight of boolean functions: $\mathbf{W}^{1}[f] = \sum_{i=1}^n \widehat{f}(i)^2.$

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## §2.3: Total influence

A very important quantity in the analysis of a boolean function is the sum of its influences.

Definition 26 The total influence of $f : \{-1,1\}^n \to {\mathbb R}$ is defined to be $\mathbf{I}[f] = \sum_{i=1}^n \mathbf{Inf}_i[f].$

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