## §7.1: Dictator testing

In Chapter 1.6 we described the BLR property testing algorithm: given query access to an unknown function $f : \{0,1\}^n \to \{0,1\}$, this algorithm queries $f$ on a few random inputs and approximately determines whether $f$ has the property of being linear over ${\mathbb F}_2$. The field of property testing for boolean functions is concerned [...]

## §6.5: Highlight: Fooling ${\mathbb F}_2$-polynomials

Recall that a density $\varphi$ is said to be $\epsilon$-biased if its correlation with every ${\mathbb F}_2$-linear function $f$ is at most $\epsilon$ in magnitude. In the lingo of pseudorandomness, one says that $\varphi$ fools the class of ${\mathbb F}_2$-linear functions:

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## §6.4: Applications in learning and testing

In this section we describe some applications of our study of pseudorandomness.

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## §6.3: Constructions of various pseudorandom functions

In this section we give some constructions of boolean functions with strong pseudorandomness properties.

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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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## §5.3: The Fourier coefficients of Majority

In this section we will analyze the Fourier coefficients of $\mathrm{Maj}_n$. In fact, we give an explicit formula for them in Theorem 16 below. But most of the time this formula is not too useful; instead, it’s better to understand the Fourier coefficients of $\mathrm{Maj}_n$ asymptotically as $n \to \infty$.

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## §5.2: Majority, and the Central Limit Theorem

Majority is one of the more important functions in boolean analysis and its study motivates the introduction of one of the more important tools: the Central Limit Theorem (CLT).

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