Abelian groups are polynomially stable
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2018
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| _version_ | 1866911949526138880 |
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| author | Becker, Oren Mosheiff, Jonathan |
| author_facet | Becker, Oren Mosheiff, Jonathan |
| contents | 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 $\operatorname{Sym}(n)$. In particular, this means that there exists $D\geq 1$ such that for $A,B\in \operatorname{Sym}(n)$, if $AB$ is $δ$-close to $BA$, then $A$ and $B$ are $ε$-close to a commuting pair of permutations, where $ε\leq O(δ^{1/D})$. We also observe a property-testing reformulation of this result, yielding efficient testers for certain permutation properties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1811_00578 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Abelian groups are polynomially stable Becker, Oren Mosheiff, Jonathan Group Theory Combinatorics 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 $\operatorname{Sym}(n)$. In particular, this means that there exists $D\geq 1$ such that for $A,B\in \operatorname{Sym}(n)$, if $AB$ is $δ$-close to $BA$, then $A$ and $B$ are $ε$-close to a commuting pair of permutations, where $ε\leq O(δ^{1/D})$. We also observe a property-testing reformulation of this result, yielding efficient testers for certain permutation properties. |
| title | Abelian groups are polynomially stable |
| topic | Group Theory Combinatorics |
| url | https://arxiv.org/abs/1811.00578 |