Note on a problem of Kátai (Q1063024)

Let \(f: {\mathbb{N}}\to {\mathbb{R}}\) be an additive function. Let E be a subset of \({\mathbb{N}}\) with the following property. If f(E)\(\subseteq {\mathbb{Z}}\) then f(\({\mathbb{N}})\subseteq {\mathbb{Z}}\). By a short group-theoretic argument it is shown that every \(n\in {\mathbb{N}}\) can be expressed in the form \(n=\prod^{k}_{i=1}e_ i^{\alpha_ i}\), \(e_ i\in E\), \(\alpha_ i\in {\mathbb{Z}}\). This answers a question of Kátai.