文摘
We prove that the multiplicative group generated by \({\{ \frac{[\sqrt{2}n]}{n} | n \in \mathbb{N} \}}\) is the group of positive rational numbers. It is proved that if a completely additive function f satisfying f \({([\sqrt{2}n]) - f(n) \rightarrow C (n \rightarrow \infty)}\) for some real number C, then \({f(n) = A{\text{log}} n}\) , where \({A = \frac{2C}{log 2}}\) .