Remarks on Farah's theorems
type: Item
instance of: scholarly article
title: Remarks on Farah's theorems
zbMATH Open document ID: 1385.39007
published in: Results in Mathematics
publication date: 2018-01-30T00:00:00Z
review text: \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.
reviewed by: Maryam Amyari
Mathematics Subject Classification ID: 39B82
Mathematics Subject Classification ID: 28A35
Mathematics Subject Classification ID: 20K30
Mathematics Subject Classification ID: 39B52
zbMATH DE Number: 6832354
zbMATH Keywords: approximate group homomorphism
zbMATH Keywords: Ulam's type stability
zbMATH Keywords: Farah's problem
author: Barbara Przebieracz
Wikidata QID: Q59608480
MaRDI profile type: Publication
full work available at URL: s00025-017-0722-4
OpenAlex ID: W2737283917
cites work: Approximate homomorphisms. II: Group homomorphisms
cites work: Hyers-Ulam stability of functional equations in several variables
cites work: Q5511350
DOI: 10.1007/S00025-017-0722-4
description: scientific article; zbMATH DE number 6832354