Boolean functions on $S_n$ which are nearly linear

TL;DR

This paper characterizes functions on the symmetric group that are nearly linear, showing they are close to unions of disjoint cosets, thus extending Friedgut-Kalai-Naor type theorems to $S_n$ with sharp bounds.

Contribution

It establishes sharp stability results for Boolean and real-valued functions on $S_n$ that are close to linear, linking them to unions of cosets, and extends fundamental theorems in Boolean analysis.

Findings

Functions close to linear are near unions of disjoint cosets.

Sharp bounds are proven for functions close to Boolean and linear functions.

Results extend Friedgut-Kalai-Naor theorem to the symmetric group.

Abstract

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.