Mean value theorems in multiplicative systems and joint ergodicity of additive and multiplicative actions

Abstract: In this paper we are concerned with the study of additive ergodic averages in multiplicative systems and the investigation of the "pretentious" dynamical behaviour of these systems. We prove a mean ergodic theorem (Theorem A) that generalises Hal\'asz's mean value theorem for finitely generated multiplicative functions taking values in the unit circle. In addition, we obtain two structural results concerning the "pretentious" dynamical behaviour of finitely generated multiplicative systems. Moreover, motivated by the independence principle between additive and multiplicative structures of the integers, we explore the joint ergodicity (as a natural notion of independence) of an additive and a finitely generated multiplicative action, both acting on the same probability space. In Theorem B, we show that such actions are jointly ergodic whenever no "local obstructions" arise, and we give a concrete description of these "local obstructions". As an application, we obtain some new combinatorial results regarding arithmetic configurations in large sets of integers including refinements of a special case of Szemer\'edi's theorem.