By Ryan O'Donnell

Boolean features are might be the main simple gadgets of analysis in theoretical laptop technology. additionally they come up in different components of arithmetic, together with combinatorics, statistical physics, and mathematical social selection. the sphere of research of Boolean capabilities seeks to appreciate them through their Fourier rework and different analytic equipment. this article offers an intensive review of the sector, starting with the main simple definitions and continuing to complex issues similar to hypercontractivity and isoperimetry. each one bankruptcy incorporates a "highlight program" comparable to Arrow's theorem from economics, the Goldreich-Levin set of rules from cryptography/learning concept, Håstad's NP-hardness of approximation effects, and "sharp threshold" theorems for random graph houses. The booklet contains approximately 450 routines and will be used because the foundation of a one-semester graduate direction. it's going to attract complicated undergraduates, graduate scholars, and researchers in laptop technological know-how thought and comparable mathematical fields.

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20 1 Boolean Functions and the Fourier Expansion (a) Show that deg(f ) = deg(a + bf ) for any a, b ∈ R (assuming b = 0, a + bf = 0). (b) Show that deg(f ) ≤ k if and only if f is a real linear combination of functions g1 , . . , gs , each of which depends on at most k input coordinates. 1 have “nontrivial” degree? 11 Suppose that f : {−1, 1}n → {−1, 1} has deg(f ) = k ≥ 1. 9), call it q(x), also has deg(q) = k. 9(c),(d), deduce that f ’s Fourier spectrum is “21−k -granular”, meaning each f (S) is an integer multiple of 21−k .

3. Total Influence 35 Let’s now take a look at more analytic expressions for the total influence. By definition, if f : {−1, 1}n → R, then n n I[f ] = Inf i [f ] = n E[Di f (x) ] = E 2 i=1 i=1 x x Di f (x)2 . 34. The (discrete) gradient operator ∇ maps the function f : {−1, 1}n → R to the function ∇f : {−1, 1}n → Rn defined by ∇f (x) = (D1 f (x), D2 f (x), . . , Dn f (x)). Note that for f : {−1, 1}n → {−1, 1} we have ∇f (x) 22 = sensf (x), where · 2 is the usual Euclidean norm in Rn . 35. For f : {−1, 1}n → R, I[f ] = E[ ∇f (x) 22 ].

4. , f (x) = g(xi1 , . . , xik ) for some g : {−1, 1}k → {−1, 1} and i1 , . . , ik ∈ [n]. Informally, we say that f is a “junta” if it depends on only a “constant” number of coordinates. For example, the number of functions f : {−1, 1}n → {−1, 1} which are 1-juntas is precisely 2n + 2: the n dictators, the n negated-dictators, and the 2 constant functions ±1. 5. A function f : {−1, 1}n → {−1, 1} is called a weighted majority or (linear) threshold function if it is expressible as f (x) = sgn(a0 + a1 x1 + · · · + an xn ) for some a0 , a1 , .