On the distribution of the fourier spectrum of Boolean functions

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The Unique Games Conjecture, Integrality Gap for Cut Problems and Embeddability of Negative Type Metrics into 1 (Extended Abstract) (2005, background): “by a computer search), and Goemans (1.024 for a metric based on the Leech Lattice), see [ 29 ].”
The unique games conjecture, integrality gap for cut problems and embeddability of negative type metrics into l/sub 1/ (2005, background): “THEOREM 2.9 (MAIN THEOREM).”
Inapproximability of NP-complete Problems, Discrete Fourier Analysis, and Geometry (2011, unknown): “(Bourgain [7]) Let c > 12 be fixed.”
Optimal inapproximability results for MAX-CUT and other 2-variable CSPs? (2004, background): “This is because S (f) is a just a weighted sum of the squared Fourier coefficients of f , where coefficients at level k have weight k. Some strong evidence in favor of the Majority Is Stablest theorem was given by Bourgain [ 7 ], who showed that non-junta functions f have their Fourier tails P jSj>k”
Intersecting Families of Permutations (2010, background): “Examples of this can be found for example in [5], [11] and [12].”