On the multiplicative group generated by $${\{ \frac{[\sqrt{2}n]}{n} | n \in \mathbb{N} \}}$$ { [ 2 n ] n | n ∈ N }

Abstract

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 $${A = \frac{2C}{log 2}}$$ .