Low Influence Functions over Slices of the Boolean Hypercube Depend on Few Coordinates

An Orthogonal Basis for Functions over a Slice of the Boolean Hypercube

“…We start with a triangle inequality for influences (cf. [9,Lemma 5.4] for the Boolean case, in which the constant 9 2 can be improved to 3 2 ).…”

“…[ Our goal in the rest of this section is to give a formula for Inf m [f ]. Our treatment closely follows Wimmer [9]. We start with a formula for f (m m+1) .…”

“…Wimmer [9] recently generalized a fundamental theorem of Friedgut [3] from the Boolean hypercube to the slice. Although Wimmer's main theorem is a statement about functions on the slice, Wimmer lifts the given function to the symmetric group, where most of his argument takes place, exploiting essential properties of Young's orthogonal representation.…”

Invariance Principle on the Slice

“…[33, Proposition 5.3],[13, Lemma 5.2]). Let f :`r ns pn˘Ñ R. For every τ ą 0 there is a set J Ď rns of size OpInf s rf s{τ q such that Inf s ij rf s ă τ whenever i, j R J.Combining Lemma 8.2 and Lemma 8.3, we deduce that bounded degree functions depend on a constant number of coordinates, the analog of [27, Theorem 1].…”

Direct Sum Testing

“…There has recently been an active line of research in the field of analysis of boolean functions extending known results regarding functions defined on the hypercube to their analogues defined on a fixed layer of the hypercube [29,34,17]. In this paper we prove an analogous result for linearity test, which is another classical result in this area.…”