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FKN theorem for the multislice, with applications

Part of: Harmonic analysis in several variables Circuits, networks Inequalities

Published online by Cambridge University Press:  18 October 2019

Yuval Filmus
Yuval Filmus*
Affiliation:
Computer Science Department, Technion – Israel Institute of Technology
*
Email: [email protected]
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Abstract

The Friedgut–Kalai–Naor (FKN) theorem states that if ƒ is a Boolean function on the Boolean cube which is close to degree one, then ƒ 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.

MSC classification

Primary: 26D07: Inequalities involving other types of functions
Secondary: 42B10: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 94C10: Switching theory, application of Boolean algebra; Boolean functions
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 29 , Issue 2 , March 2020 , pp. 200 - 212
Copyright
© Cambridge University Press 2019

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Footnotes

†

Taub Fellow, supported by the Taub Foundations. The research was funded by ISF grant 1337/16.

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