On approximate group homomorphisms
type: Item
instance of: scholarly article
title: On approximate group homomorphisms
zbMATH Open document ID: 1390.39097
author: Roman Badora
author: Barbara Przebieracz
published in: Journal of Mathematical Analysis and Applications
publication date: 2018-05-31T00:00:00Z
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. After that \textit{B. Przebieracz} [Result. Math. 72, No. 4, 1959--1966 (2017; Zbl 1385.39007)] presented the versions of Farah's results without extra assumptions but \(G\) is assumed to be finite. In this paper, the authors prove the following statement: let \(G\) be a group (not necessarily finite) with a right-invariant, finitely additive probability measure \(\mu : P(G)\to [0,1]\), \(H\) be a group with an invariant metric \(d:H\times H \to [0, \infty), \varepsilon \geq 0\), and let \(0\leq \delta <\frac{1}{12}\). If a function \(f:G\to H\) satisfies \(\mu^2(\{(x,y):\,\,\,d(f(xy), f(x)f(y))>\varepsilon \}\leq \delta,\) then for every \(\zeta > 0\) there exists a map \(F:G\to H\) such that \(d(F(xy), F(x)F(y))\leq 24\varepsilon,\,\,\, x, y \in G\) and \(\mu (\{x\in G: d(f(x),F(x))>\varepsilon\})\leq 4\delta +\zeta.\)
reviewed by: Maryam Amyari
Mathematics Subject Classification ID: 39B82
Mathematics Subject Classification ID: 20K30
Mathematics Subject Classification ID: 39B52
zbMATH DE Number: 6877138
zbMATH Keywords: approximate group homomorphism
zbMATH Keywords: Ulam's stability problem
zbMATH Keywords: Erdős' problem
zbMATH Keywords: Farah's problem
MaRDI profile type: Publication
full work available at URL: j.jmaa.2018.02.017
OpenAlex ID: W2791795169
cites work: More on Hyers' theorem
cites work: Amenable semigroups
cites work: On almost additive functions
cites work: Approximate homomorphisms. II: Group homomorphisms
cites work: The stability of homomorphisms and amenability, with applications to functional equations
cites work: On stability of the Cauchy equation on semigroups
cites work: Note on almost additive functions
cites work: Q3669878
cites work: On the Stability of the Linear Functional Equation
cites work: On the asymptoticity aspect of Hyers-Ulam stability of mappings
cites work: On Cauchy's Functional Equation
cites work: Invariant measures
cites work: Q3756753
cites work: Remarks on Farah's theorems
cites work: Q3742042
cites work: Proper families and almost additive functions
cites work: Stability of the Cauchy equation almost everywhere
cites work: Q3259049
DOI: 10.1016/J.JMAA.2018.02.017
description: scientific article; zbMATH DE number 6877138