ON A -ADDITIVE UNIQUENESS SET FOR MULTIPLICATIVE FUNCTIONS

Abstract Let $k\geq 2$ be an integer. We prove that the 2-automatic sequence of odious numbers $\mathcal {O}$ is a k -additive uniqueness set for multiplicative functions: if function f satisfies multivariate Cauchy’s functional equation $f(x_1+x_2+\cdots +x_k)=f(x_1)+f(x_2)+\cdots +f(x_k)$ arbitrary $x_1,\ldots ,x_k\in \mathcal , then identity $f(n)=n$ all $n\in \mathbb {N}$ .