Boolean Functions on Sn Which Are Nearly Linear

Yuval Filmus

doi:10.19086/da.30186

Boolean Functions on Sn Which Are Nearly Linear

Yuval Filmus

Computer Science

Research output: Contribution to journal › Article › peer-review

Abstract

We show that if (Formula Presented) is e-close to linear in L2 and (Formula Presented) then f is O(e)-close to a union of “mostly disjoint” cosets, and moreover this is sharp: any such union is close to linear. This constitutes a sharp Friedgut–Kalai–Naor theorem for the symmetric group. Using similar techniques, we show that if (Formula Presented) is linear, (Formula Presented), and (Formula Presented), then (Formula Presented)-close to a union of mostly disjoint cosets, and this is also sharp; and that if f: Sn → R is linear and e-close to f0;1g in L∞ then f is O(ε)-close in L∞ to a union of disjoint cosets.

Original language	English
Article number	25
Journal	Discrete Analysis
Volume	2021
DOIs	https://doi.org/10.19086/da.30186
State	Published - 2021

Keywords

Analysis of boolean functions
Symmetric group

ASJC Scopus subject areas

Algebra and Number Theory
Geometry and Topology
Discrete Mathematics and Combinatorics

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10.19086/da.30186

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AB - We show that if (Formula Presented) is e-close to linear in L2 and (Formula Presented) then f is O(e)-close to a union of “mostly disjoint” cosets, and moreover this is sharp: any such union is close to linear. This constitutes a sharp Friedgut–Kalai–Naor theorem for the symmetric group. Using similar techniques, we show that if (Formula Presented) is linear, (Formula Presented), and (Formula Presented), then (Formula Presented)-close to a union of mostly disjoint cosets, and this is also sharp; and that if f: Sn → R is linear and e-close to f0;1g in L∞ then f is O(ε)-close in L∞ to a union of disjoint cosets.

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Boolean Functions on Sn Which Are Nearly Linear

Abstract

Keywords

ASJC Scopus subject areas

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