On the Orbits of Multiplicative Pairs

TL;DR

This paper characterizes pairs of multiplicative functions with orbits that do not densely fill the torus, settling a longstanding conjecture in the field.

Contribution

It provides a complete characterization of such pairs, resolving an old conjecture by Daróczy and Kátai.

Findings

Identifies all pairs of multiplicative functions with non-dense orbits

Settles the conjecture of Daróczy and Kátai

Advances understanding of multiplicative function dynamics

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 Zolt\'an Dar\'oczy and Imre K\'atai.