
Some tips
 You might try using analysis of Boolean functions whenever you’re faced with a problems involving Boolean strings in which both the uniform probability distribution and the Hamming graph structure play a role. More generally, the tools may still apply when studying functions on (or subsets of) product probability spaces.
 If you’re mainly interested in unbiased functions, or subsets of volume $\frac12$, use the representation $f : \{1,1\}^n \to \{1,1\}$. If you’re mainly interested in subsets of small volume, use the representation $f : \{1,1\}^n \to \{0,1\}$.
 As for the domain, if you’re interested in the operation of adding two strings (modulo $2$), use $\mathbb{F}_2^n$. Otherwise use $\{1,1\}^n$.
 If you have a conjecture about Boolean functions:
 Test it on dictators, majority, parity, tribes (and maybe recursive majority of $3$). If it’s true for these functions, it’s probably true.
 Try to prove it by induction on $n$.
 Try to prove it in the special case of functions on Gaussian space.
 Try not to prove any bound on Boolean functions $f : \{1,1\}^n \to \{1,1\}$ that involves the parameter $n$.
 Analytically, the only multivariate polynomials we really know how to control are degree$1$ polynomials. Try to reduce to this case if you can.
 Hypercontractivity is useful in two ways: (i) It lets you show that lowdegree functions of independent random variables behave “reasonably”. (ii) It implies that the noisy hypercube graph is a smallset expander.
 Almost any result about functions on the hypercube extends to the case of the $p$biased cube, and more generally, to the case of functions on products of discrete probability spaces in which every outcome has probability at least $p$ — possibly with a dependence on $p$, though.
 Every Boolean function consists of a junta part and Gaussian part.


Thanks for the tips! In the third tip (about the domain), should $\{ 1, 1 \}$ be $\{ 1, 1 \}^n$? Can you say a word about the rationale for the fifth tip (about the parameter $n$)?
You’re right about the third tip, thanks!
Regarding the 5th… well of course it’s not ironclad, but one of the themes in the area is to try to work in “$\{1,1\}^\infty$” if possible. I guess I’m mainly thinking about theorems like hypercontractivity and the like (ones proven by induction) where one might be tempted to prove weaker versions with a dependence on $n$, but where it’s possible to prove something independent of $n$.
Thanks for the explanation. A million thanks for the entire book!
Is there an “nfree” variant (generalization?) of KKL that has more or less the same applications?
Luca: Yes, take a look at the “KKL EdgeIsoperimetric Theorem” in Chapter 9.6, which gives what I view as the “correct” version of the KKL theorem.