Abstract

We prove a pointwise convergence result for additive ergodic averages associated with certain multiplicative actions of the Gaussian integers. We derive several applications in dynamics and number theory, including: (i) Wirsing’s theorem for Gaussian integers: if f : G ? R is a bounded completely multiplicative function, then the following limit exists:(Formula Presented) (ii) An answer to a special case of a question of Frantzikinakis and Host: for any completely multiplicative real-valued function f : N ? R, the following limit exists: (Formula Presented) (iii) A variant of a theorem of Bergelson and Richter on ergodic averages along the ? function: if (X, T) is a uniquely ergodic system with unique invariant measure ?, then for any x ? X and f ? C(X), (Formula Presented) © 2024 American Mathematical Society.