On the characterization of additive functions with monotonic norm (Q1078605)

The author proves an interesting generalization of a well-known theorem of \textit{P. Erdős} [Ann. Math., II. Ser. 47, 1-20 (1946; Zbl 0061.079)]: If a function \(f: {\mathbb{N}}\to {\mathbb{R}}^ k\) is additive and its Euclidean norm is monotonic from some point on, then \(f(n)=c \log n\). The proof is elementary. The remarks contain some problems about the generalization of the theorem to finite dimensional vector spaces with different norm and for Hilbert spaces.