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 $f\colon \mathbb{ G} \to \mathbb{ R}$ is a bounded completely multiplicative function, then the following limit exists: $$\lim_{ N \to \infty} \frac{ 1}{ N^2} \sum_{ 1 \leq m, n \leq N} f(m + { \rm i} n).$$ (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: $$\lim_{ N \to \infty} \frac{ 1}{ N^2} \sum_{ 1 \leq m, n \leq N} f(m^2 + n^2).$$ (iii) A variant of a theorem of Bergelson and Richter on ergodic averages along the $\Omega$ function: if $(X,T)$ is a uniquely ergodic system with unique invariant measure $\mu$, then for any $x\in X$ and $f\in C(X)$, $$\lim_{ N\to\infty}\frac{ 1}{ N^2}\sum_{ 1 \leq m, n \leq N} f(T^{ \Omega(m^2 + n^2)}x)=\int_Xf \ d\mu.$$
Categories: math.DS