Remarks on Farah's theorems
\textit{I. Farah} [Combinatorica 20, No. 1, 47--60 (2000; Zbl 1008.28004)] investigated mappings between finite product groups which are approximately homomorphisms with respect to some metric on the range. In the present paper, the author presents the versions of Farah's results without extra assumptions as follows: Let \((G, +)\) be a finite abelian group, \(\varepsilon \geq 0\). Let \((H, +)\) be an abelian group with translation-invariant metric \(d: H\times H \to [0,\infty),\) let \(f : G \to H\), and let \(\delta \in (0, 1-\frac{\sqrt{3}}{2})\). Suppose that \[ \mu (\{(x, y): d (f(x) + f(y), f(x + y)) > \varepsilon \}) \leq \delta. \] The author shows that there is a \(20\varepsilon\)-approximate homomorphism \(F : G \to H\) (i.e., \(d (F(a + b), F(a) + F(b)) \leq 20\varepsilon\) for all \( a, b \in G\)) such that \( \nu (\{x : d (f(x), F(x)) > 7\varepsilon\}) \leq \frac{\delta}{1- 2\delta}\), where \(\mu\) and \(\nu\) are special measures on \(2^{G\times G}\) and \(2^G\) into \([0,1]\), respectively.
scientific article; zbMATH DE number 6832354