Friedgut-Kalai-Naor Theorem for Slices of the Boolean Cube

The Friedgut–Kalai–Naor theorem, a basic result in the field of analysis of Boolean functions, states that if a Boolean function on the Boolean cube {0,1}n is close to a function of the form c0 +∑i cixi, then it is close to a dictatorship (a function depending on a single coordinate). We prove an analogous theorem for functions defined on the slice ([n] k ) = {(x1, . . . ,xn) ∈ {0,1}n : ∑i xi = k}. When k/n is bounded away from 0 and 1, our theorem states that if a function on the slice is close to a function of the form ∑i cixi then it is close to a dictatorship. When k/n is close to 0 or to 1, we can only guarantee being close to a junta (a function depending on a small number of coordinates); this deterioration in the guarantee is unavoidable, since for small p a maximum of a small number of variables is close to their sum. Kindler and Safra proved an FKN theorem for the biased Boolean cube, in which the underlying measure is the product measure μp(x) = p∑i xi(1− p)∑i(1−xi). As a corollary of our FKN theorem for the slice, we deduce a uniform version of the FKN theorem for the biased Boolean cube, in which the error bounds depend uniformly on p. Mirroring the situation on the slice, when p is very close to 0 or to 1, we can only guarantee closeness to a junta. *Research conducted at the Simons Institute for the Theory of Computing during the 2013 fall semester on Real Analysis in Computer Science, and at the Institute for Advanced Study, Princeton, NJ. This material is based upon work supported by the National Science Foundation under agreement No. DMS-1128155. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors, and do not necessarily reflect the views of the National Science Foundation.