Almost linear Boolean functions on S_n are almost unions of cosets

07/16/2021
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by   Yuval Filmus, et al.
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We show that if f S_n β†’{0,1} is Ο΅-close to linear in L_2 and 𝔼[f] ≀ 1/2 then f is O(Ο΅)-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 βˆ‰{0,1}] ≀ϡ, and [f = 1] ≀ 1/2, then f is O(Ο΅)-close to a union of mostly disjoint cosets, and this is also sharp; and that if f S_n →ℝ is linear and Ο΅-close to {0,1} in L_∞ then f is O(Ο΅)-close in L_∞ to a union of disjoint cosets.

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