On the Orbits of Multiplicative Pairs

Abstract

We characterize all pairs of completely multiplicative functions $f,g:\mathbb{N}\to \mathbb{T}$ such that the orbit closure \[\overline{\{(f(n),g(n+1))\}_{n\ge 1}} \neq \mathbb{T}\times \mathbb{T}.\] In so doing, we settle an old conjecture of Zoltán Daróczy and Imre Kátai.

30 pages