Multiplicative functions which are additive on some automatic sequences

Abstract. Let \(k\geq2\) be an integer. We prove that the \(2\)-automatic sequence \(\mathcal{E}\) of numbers with an even number of zeros in their base \(2\) representation is a \(k\)-additive uniqueness set for the set of multiplicative functions. That is, if a multiplicative function \(f_k\) satisfies a multivariate Cauchy’s functional equation \[ f_k(x_1+x_2+\cdots+x_k)=f_k(x_1)+f_k(x_2)+\cdots+f_k(x_k) \] for arbitrary \(x_1,\ldots,x_k\in \mathcal{E}\), then \(f_k\) is the identity function \(f_k(n)=n\) for all \(n\in\mathbb{N}\). We also show that the Fibonacci-automatic sequence \(\mathcal{F}\) of numbers with an odd number of 1's in their Zeckendorf representation is a \(k\)-additive uniqueness set for the set of multiplicative functions.

Key words and phrases. Additive uniqueness, multiplicative functions, automatic sequences.