On Ulam stability of the real line (Q2772751)

By an approximate homomorphism of groups \(A\), \(B\) is meant a mapping \(f:A\to B\) such that \(f(xy)\) is, in some sense, close to \(f(x)f(y)\). The stability of a category of approximate homomorphisms then means a possibility to approximate approximate homomorphisms by strict homomorphisms. In the case of metric ``close'' can mean ``to be at a small distance'' and stability can mean that every \(\varepsilon\)-approximate homomorphism can be \(K\varepsilon\)-approximated by a strict homomorphism with a fixed constant \(K\). In the paper the author considers an algebraic version of stability for homomorphisms of \(\mathbb R\) by treating \(x\), \(y\) to be ``close'' when the difference \(x-y\) belongs to a fixed subgroup \(G\subseteq\mathbb R\). He proves two theorems: (1) If \(G\subseteq\mathbb R\) is a countable group and \(f:\mathbb R\to\mathbb R\) is a Borel \(G\)-approximate homomorphism then there is a real \(r\) such that \(f(x)-rx\in G\) for all \(x\). (2) If \(\oplus\) is a Borel binary operation of \(\mathbb R\) such that \(\langle\mathbb R,{\oplus}\rangle\) is an abelian group and the difference \((x\oplus y)-(x+y)\) takes only countably many values then \(\langle\mathbb R,{\oplus}\rangle\) is Borel isomorphic to \(\langle\mathbb R,{+}\rangle\).NEWLINENEWLINEFor the entire collection see [Zbl 0972.00067].