Yuval Filmus

Chicago Journal of Theoretical Computer Science

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.

@article{Filmus2016b,

author = {Yuval Filmus},

title = {Friedgut--{K}alai--{N}aor theorem for slices of the {B}oolean cube},

journal = {Chicago Journal of Theoretical Computer Science},

year = {2016},

pages = {14:1--14:17}

}

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author = {Yuval Filmus},

title = {Friedgut--{K}alai--{N}aor theorem for slices of the {B}oolean cube},

journal = {Chicago Journal of Theoretical Computer Science},

year = {2016},

pages = {14:1--14:17}

}