Boolean functions on S_n which are nearly linear

We show that if (Formula Presented) is e-close to linear in L2 and (Formula Presented) then f is O(e)-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 (Formula Presented) is linear, (Formula Presented), and (Formula Presented), then (Formula Presented)-close to a union of mostly disjoint cosets, and this is also sharp; and that if f: Sn → R is linear and e-close to f0;1g in L∞ then f is O(ε)-close in L∞ to a union of disjoint cosets.