Additive functions (Q1821135)

[This article is reviewed together with the following.] Let \(1<q<2\) and \(L=1/(q-1)\). Every \(x\in [0,L]\) has a (generally not unique) representation in the form \(x=\sum^{\infty}_{n=1}\epsilon_ nq^{-n}\), \(\epsilon_ n=0\) or 1. The authors select one of them (the one given by a greedy algorithm) and call it the regular expansion of x. They call a function F: [0,L]\(\to {\mathbb{R}}\) additive, if it satisfies (*) \(F(x)=\sum \epsilon_ nF(q^{-n})\) for the regular expansion of x; completely additive, if (*) holds for all possible expansions of x with \(\epsilon_ n\in \{0,1\}\). The authors investigate the possible additive functions under various restrictions like continuity, differentiability, boundedness. Among others, they find that for \(q=(\sqrt{5}+1)/2\) every continuous additive function is linear and that, for every q, if a continuous additive function is either nonnegative or differentiable at any point, then it must be linear. On the other hand, for certain values of q there exist continuous additive functions that are not completely additive (and consequently are nowhere differentiable).