Junta threshold for low degree Boolean functions on the slice

Abstract

We show that a Boolean degree $d$ function on the slice $\left(\genfrac{}{}{0ex}{}{[n]}{k}\right)$ is a junta if $k\ge 2d$, and that this bound is sharp. We prove a similar result for $A$-valued degree $d$ functions for arbitrary finite $A$, and for functions on an infinite analog of the slice.

16 pages