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The Friedgut–Kalai–Naor theorem is a fundamental result in the analysis of Boolean function. It states that if a Boolean function $f$ is close to an affine function, then $f$ is close to an affine Boolean function, which must depend on at most one coordinate. We prove an analog of this theorem for slices of the Boolean cube (a slice consists of all vectors having a given Hamming weight). In the small error regime, our theorem shows that $f$ is close to a function depending on at most one coordinate, and in general we show that $f$ or its negation is close to a maximum of a small number of coordinates (this corresponds to a union of stars, families consisting of all elements containing some fixed element).

See also our later simplified account.