Boolean functions on $S_n$ which are nearly linear

Abstract

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

27 pages

References in corpus (1)

• A tighter bound on the number of relevant variables in a bounded degree Boolean function