§9.1: Low-degree polynomials are reasonable

As anyone who has worked in probability knows, a random variable can sometimes behave in rather “unreasonable” ways. It may be never close to its expectation. It might exceed its expectation almost always, or almost never. It might have finite $1$st, $2$nd, and $3$rd moments, but an infinite $4$th moment. All of this poor behaviour [...]

Chapter 9: Basics of hypercontractivity

In 1970, Bonami proved the following central result:

The Hypercontractivity Theorem Let $f : \{-1,1\}^n \to {\mathbb R}$ and let ${1 \leq p \leq q \leq \infty}$. Then $\|\mathrm{T}_\rho f\|_q \leq \|f\|_p$ for $0 \leq \rho \leq \sqrt{\tfrac{p-1}{q-1}}$.

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Chapter 8 notes

The origins of the orthogonal decomposition described in Section 3 date back to the work of Hoeffding [Hoe48] (see also [vMis47]).

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§8.6: Highlight: Randomized decision tree complexity

A decision tree $T$ for $f : \{-1,1\}^n \to \{-1,1\}$ can be thought of as a deterministic algorithm which, given adaptive query access to the bits of an unknown string $x \in \{-1,1\}^n$, outputs $f(x)$. E.g., to describe a natural decision tree for $f = \mathrm{Maj}_3$ in words: “Query $x_1$, then $x_2$. If they [...]

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