A group version of stable regularity. (English) Zbl 1539.03116
Summary: We prove that, given \(\epsilon > 0\) and \(k \geq 1\), there is an integer \(n\) such that the following holds. Suppose \(G\) is a finite group and \(A \subseteq G\) is \(k\)-stable. Then there is a normal subgroup \(H \leq G\) of index at most \(n\), and a set \(Y \subseteq G\), which is a union of cosets of \(H\), such that \(|A \triangle Y| \leq \epsilon |H|\). It follows that, for any coset \(C\) of \(H\), either \(|C \cap A|\leq \epsilon|H|\) or \(|C A| \leq \epsilon |H|\). This qualitatively generalises recent work of Terry and Wolf on vector spaces over \(\mathbb{F}_p\).
MSC:
| 03C45 | Classification theory, stability, and related concepts in model theory |
| 03C60 | Model-theoretic algebra |
| 20A15 | Applications of logic to group theory |
References:
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