Boolean constant degree functions on the slice are juntas (1801.06338v2)

Abstract: We show that a Boolean degree $d$ function on the slice $\binom{[n]}{k} = { (x_1,\ldots,x_n) \in {0,1} : \sum_{i=1}^n x_i = k }$ is a junta, assuming that $k,n-k$ are large enough. This generalizes a classical result of Nisan and Szegedy on the hypercube. Moreover, we show that the maximum number of coordinates that a Boolean degree $d$ function can depend on is the same on the slice and the hypercube.