Almost linear Boolean functions on S_n are almost unions of cosets
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We show that if f S_n →{0,1} is ϵ-close to linear in L_2 and 𝔼[f] ≤ 1/2 then f is O(ϵ)-close to a union of "mostly disjoint" cosets, and moreover this is sharp: any such union is close to linear. This constitutes a sharp Friedgut-Kalai-Naor theorem for the symmetric group. Using similar techniques, we show that if f S_n →ℝ is linear, [f ∉{0,1}] ≤ϵ, and [f = 1] ≤ 1/2, then f is O(ϵ)-close to a union of mostly disjoint cosets, and this is also sharp; and that if f S_n →ℝ is linear and ϵ-close to {0,1} in L_∞ then f is O(ϵ)-close in L_∞ to a union of disjoint cosets.
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