By Ryan O'Donnell

Boolean capabilities are possibly the main uncomplicated items of research in theoretical laptop technological know-how. additionally they come up in different components of arithmetic, together with combinatorics, statistical physics, and mathematical social selection. the sector of study of Boolean features seeks to appreciate them through their Fourier rework and different analytic equipment. this article offers an intensive evaluate of the sector, starting with the main easy definitions and continuing to complex themes reminiscent of hypercontractivity and isoperimetry. every one bankruptcy features a "highlight program" corresponding to Arrow's theorem from economics, the Goldreich-Levin set of rules from cryptography/learning conception, Håstad's NP-hardness of approximation effects, and "sharp threshold" theorems for random graph homes. The publication comprises approximately 450 workouts and will be used because the foundation of a one-semester graduate direction. it's going to attract complex undergraduates, graduate scholars, and researchers in laptop technology concept and similar mathematical fields.

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20 Let f : {−1, 1}n → R satisfy f = f =1 . Show that Var[f 2 ] = 2 2 i=j f (i) f (j ) . 21 Prove that there are no functions f : {−1, 1}n → {−1, 1} with exactly 2 nonzero Fourier coefficients. What about exactly 3 nonzero Fourier coefficients? 26. 23 In this exercise you will prove some basic facts about “distances” between probability distributions. Let ϕ and ψ be probability densities on Fn2 . (a) Show that the total variation distance between ϕ and ψ, defined by dTV (ϕ, ψ) = maxn A⊆F2 is equal to 1 2 ϕ − ψ 1.

Unfortunately, as Condorcet himself noted, there may not be a Condorcet winner. In the example above, if voter #1’s ranking was instead c > a > b (corresponding to (+1, −1, +1)), we would obtain the “paradoxical” outcome (+1, +1, +1): society prefers a over b, b over c, and c over a! This lack of a Condorcet winner is termed Condorcet’s Paradox; it occurs when the outcome (f (x), f (y), f (z)) is one of the two “all-equal” triples {(−1, −1, −1), (+1, +1, +1)}. 5. Highlight: Arrow’s Theorem 43 One might wonder if the Condorcet Paradox can be avoided by using a voting rule f : {−1, 1}n → {−1, 1} other than majority.

9). , f is -close to the linear function χS∗ . In fact, for small one can show that f is more like ( /3)-close to linear, and this is sharp. 28. The BLR Test shows that given black-box access to f : Fn2 → {−1, 1}, we can “test” whether f is close to some linear function χS using just 3 queries. 27). Nevertheless, we can still determine the value of χS (x) with high probability for every x ∈ Fn2 of our choosing using just 2 queries. This property is called local correctability of linear functions.