Junta Threshold for Low Degree Boolean Functions on the Slice

Abstract

We show that a Boolean degree~$d$ function on the slice $\binom{[n]}{k}$ is a junta if $k \geq 2d$, 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.