Abelian Groups Are Polynomially Stable

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.