Averages of completely multiplicative functions over the Gaussian integers -- a dynamical approach

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 $f\text{\hspace{0.17em}⁣}:\mathbb{G}\to \mathbb{R}$ is a bounded completely multiplicative function, then the following limit exists: $$\underset{N\to \mathrm{\infty }}{\lim }\frac{1}{N^2}\sum _{1\le m,n\le N}f(m+\mathrm{i}n)\mathrm{.}$$ (ii) An answer to a special case of a question of Frantzikinakis and Host: for any completely multiplicative real-valued function $f:\mathbb{N}\to \mathbb{R}$, the following limit exists: $$\underset{N\to \mathrm{\infty }}{\lim }\frac{1}{N^2}\sum _{1\le m,n\le N}f(m^2+n^2)\mathrm{.}$$ (iii) A variant of a theorem of Bergelson and Richter on ergodic averages along the $\mathrm{\Omega }$ function: if $(X,T)$ is a uniquely ergodic system with unique invariant measure $\mu$, then for any $x\in X$ and $f\in C(X)$, $$\underset{N\to \mathrm{\infty }}{\lim }\frac{1}{N^2}\sum _{1\le m,n\le N}f(T^{\mathrm{\Omega }(m^2+n^2)}x)={\int }_{X}fd\mu \mathrm{.}$$