On additive functions with respect to the expansion of real numbers into generalized number systems (Q2714143)

Let \(N\neq 0,\pm 1\) be an integer, \(\mathbb A=\{a_0=0,a_1,\dots,a_{|N|-1}\}\) a complete residue system \(\mod N\). A system \((\mathbb A, N)\) is said to be a number system if each integer \(n\) can be written in finite form as \(n=b_0+b_1N+\dots+b_kN^k\), \(b_i\in \mathbb A\). A function \(F\) is called additive (with respect to \((\mathbb A,N))\) if \(F(0)=0\) and for each \(x=\sum^k_{j=-\infty}x_j N^j\) (where \(x_j\) are taken from \(\mathbb A\)) and \(F(x)=\sum^k_{j=-\infty}F(x_jN^j)\), \(\sum^k_{j=-\infty}|F(x_j N^j)|<\infty\). The authors study connections between number systems and additive functions. They prove the following theorem. If \((\mathbb A,N)\) is a number system and \(F\) an additive function then \(F(x)=cx\) for \(x\in R\) (where \(R\) denotes the set of real numbers).