FKN theorem for the multislice, with applications

TL;DR

This paper extends the Friedgut-Kalai-Naor theorem from the Boolean cube and the slice to the multislice, providing new stability results for edge-isoperimetric inequalities in multicoloured settings.

Contribution

The paper generalizes the FKN theorem to the multislice, a multicoloured extension of the slice, and applies it to stability results in edge-isoperimetric problems.

Findings

Extended FKN theorem to multislice setting

Proved stability of edge-isoperimetric inequality for multislice

Demonstrated dictator functions as optimal solutions in certain parameters

Abstract

The Friedgut-Kalai-Naor (FKN) theorem states that if $f$ is a Boolean function on the Boolean cube which is close to degree 1, then $f$ is close to a dictator, a function depending on a single coordinate. The author has extended the theorem to the slice, the subset of the Boolean cube consisting of all vectors with fixed Hamming weight. We extend the theorem further, to the multislice, a multicoloured version of the slice. As an application, we prove a stability version of the edge-isoperimetric inequality for settings of parameters in which the optimal set is a dictator.