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Abelian groups are polynomially stable

Oren Becker, Jonathan Mosheiff

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 $\operatorname{Sym}(n)$. 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.

Abelian groups are polynomially stable

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 . In particular, this means that there exists such that for , if is -close to , then and are -close to a commuting pair of permutations, where . We also observe a property-testing reformulation of this result, yielding efficient testers for certain permutation properties.

Paper Structure

This paper contains 22 sections, 39 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. A review of stability in permutations
  3. Quantitative stability
  4. Application to property testing
  5. Previous work
  6. The stability rate is a group invariant
  7. Stability and graphs of actions
  8. Stability in terms of group actions
  9. Graphs of actions
  10. Abelian groups are polynomially stable
  11. Proof plan
  12. The algorithm constructing $Y$ and $f$
  13. Mapping a single $\Gamma$-set $Y_{x}$ into $X$
  14. Geometric definitions
  15. A review of reduction theory (of sublattices of $\mathbb{{Z}}^{l}$)
  16. ...and 7 more sections

Key Result

Proposition 1.7

AP15 Let $E_{1}$ and $E_{2}$ be equivalent equation-sets. Then $E_{1}$ is stable if and only if $E_{2}$ is stable.

Theorems & Definitions (104)

  • Example 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Proposition 1.7
  • Definition 1.8
  • Theorem 1.9
  • Definition 1.10
  • ...and 94 more