Friedgut--Kalai--Naor theorem for slices of the Boolean cube

Abstract

The Friedgut--Kalai--Naor theorem states that if a Boolean function $f\colon \{0,1\}^n \to \{0,1\}$ is close (in $L^2$-distance) to an affine function $\ell(x_1,...,x_n) = c_0 + \sum_i c_i x_i$, then $f$ is close to a Boolean affine function (which necessarily depends on at most one coordinate). We prove a similar theorem for functions defined over $\binom{[n]}{k} = \{(x_1,...,x_n) \in \{0,1\}^n : \sum_i x_i = k \}$.