Boolean functions on $S_n$ which are nearly linear

We show that if $f\colon S_n \to \{0,1\}$ is $\epsilon$-close to linear in $L_2$ and $\mathbb{E}[f] \leq 1/2$ then $f$ is $O(\epsilon)$-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 $f\colon S_n \to \mathbb{R}$ is linear, $\Pr[f \notin \{0,1\}] \leq \epsilon$, and $\Pr[f = 1] \leq 1/2$, then $f$ is $O(\epsilon)$-close to a union of mostly disjoint cosets, and this is also sharp; and that if $f\colon S_n \to \mathbb{R}$ is linear and $\epsilon$-close to $\{0,1\}$ in $L_\infty$ then $f$ is $O(\epsilon)$-close in $L_\infty$ to a union of disjoint cosets.