TY - JOUR

T1 - Abelian Groups Are Polynomially Stable

AU - Becker, Oren

AU - Mosheiff, Jonathan

N1 - Publisher Copyright:
© 2020 The Author(s) 2020. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permission@oup.com.

PY - 2021/10/1

Y1 - 2021/10/1

N2 - In recent years, there has been a considerable amount of interest in stability of equations and their corresponding groups. Here, we initiate the systematic study of the quantitative aspect of this theory. We develop a novel method, inspired by the Ornstein-Weiss quasi-tiling technique, to prove that abelian groups are polynomially stable with respect to permutations, under the normalized Hamming metrics on the groups $\textrm{Sym}(n)$. In particular, this means that there exists $D\geq 1$ such that for $A,B\in \textrm{Sym}(n)$, if $AB$ is $\delta $-close to $BA$, then $A$ and $B$ are $\epsilon $-close to a commuting pair of permutations, where $\epsilon \leq O\left (\delta ^{1/D}\right) $. We also observe a property-testing reformulation of this result, yielding efficient testers for certain permutation properties.

AB - In recent years, there has been a considerable amount of interest in stability of equations and their corresponding groups. Here, we initiate the systematic study of the quantitative aspect of this theory. We develop a novel method, inspired by the Ornstein-Weiss quasi-tiling technique, to prove that abelian groups are polynomially stable with respect to permutations, under the normalized Hamming metrics on the groups $\textrm{Sym}(n)$. In particular, this means that there exists $D\geq 1$ such that for $A,B\in \textrm{Sym}(n)$, if $AB$ is $\delta $-close to $BA$, then $A$ and $B$ are $\epsilon $-close to a commuting pair of permutations, where $\epsilon \leq O\left (\delta ^{1/D}\right) $. We also observe a property-testing reformulation of this result, yielding efficient testers for certain permutation properties.

UR - http://www.scopus.com/inward/record.url?scp=85122210387&partnerID=8YFLogxK

U2 - 10.1093/imrn/rnaa017

DO - 10.1093/imrn/rnaa017

M3 - Article

AN - SCOPUS:85122210387

SN - 1073-7928

VL - 2021

SP - 15574

EP - 15632

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

IS - 20

ER -