Abelian Groups Are Polynomially Stable

Oren Becker, Jonathan Mosheiff

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

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.

Original languageEnglish
Pages (from-to)15574-15632
Number of pages59
JournalInternational Mathematics Research Notices
Volume2021
Issue number20
DOIs
StatePublished - 1 Oct 2021
Externally publishedYes

ASJC Scopus subject areas

  • General Mathematics

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