§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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§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 $k \in {\mathbb N}$, \begin{equation} \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. \end{equation}

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§6.1: Notions of pseudorandomness

The most obvious spectral property of a truly random function $\boldsymbol{f} : \{-1,1\}^n \to \{-1,1\}$ is that all of its Fourier coefficients are very small (as we saw in Exercise 5.8).

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§2.4: Noise stability

Suppose $f : \{-1,1\}^n \to \{-1,1\}$ is a voting rule for a $2$-candidate election. Making the impartial culture assumption, the $n$ voters independently and uniformly randomly choose their votes ${\boldsymbol{x}} = ({\boldsymbol{x}}_1, \dots, {\boldsymbol{x}}_n)$. Now imagine that when each voter goes to the ballot box there is some chance that their vote is misrecorded.

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