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

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 $f\colon \mathbb{G} \to \mathbb{R}$ 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 $Ω$ function: if $(X,T)$ is a uniquely ergodic system with unique invariant measure $μ$, then for any $x\in X$ and $f\in C(X)$, $$\lim_{N\to\infty}\frac{1}{N^2}\sum_{1 \leq m, n \leq N} f(T^{Ω(m^2 + n^2)}x)=\int_Xf \ dμ.$$

Figures (1)

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Theorems & Definitions (64)

• Theorem 1
• Theorem 1.2
• Theorem 2
• Remark 1.3
• Remark 1.4
• Theorem 3
• Theorem 1.5: Bergelson_Richter_2020
• Theorem 4
• Corollary 1.6
• Theorem 5