Junta threshold for low degree Boolean functions on the slice

TL;DR

This paper establishes a sharp threshold for Boolean degree d functions on the slice to be juntas, extending the result to A-valued functions and an infinite analog, thus advancing understanding of function structure.

Contribution

It proves that Boolean degree d functions on the slice are juntas if k ≥ 2d, and generalizes this to A-valued functions and infinite slices, with sharp bounds.

Findings

Boolean degree d functions are juntas if k ≥ 2d

The bound k ≥ 2d is sharp for the junta property

Results extend to A-valued functions and infinite slices

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.