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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2023
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| Accès en ligne: | https://arxiv.org/abs/2309.07249 |
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| _version_ | 1866914704140533760 |
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| author | Donoso, Sebastián Le, Anh N. Moreira, Joel Sun, Wenbo |
| author_facet | Donoso, Sebastián Le, Anh N. Moreira, Joel Sun, Wenbo |
| contents | 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 $f\in C(X)$, $$\lim_{N\to\infty}\frac{1}{N^2}\sum_{1 \leq m, n \leq N} f(T^{Ω(m^2 + n^2)}x)=\int_Xf \ dμ.$$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_07249 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Averages of completely multiplicative functions over the Gaussian integers -- a dynamical approach Donoso, Sebastián Le, Anh N. Moreira, Joel Sun, Wenbo Dynamical Systems Primary: 37A44 and 11N99 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 $f\in C(X)$, $$\lim_{N\to\infty}\frac{1}{N^2}\sum_{1 \leq m, n \leq N} f(T^{Ω(m^2 + n^2)}x)=\int_Xf \ dμ.$$ |
| title | Averages of completely multiplicative functions over the Gaussian integers -- a dynamical approach |
| topic | Dynamical Systems Primary: 37A44 and 11N99 |
| url | https://arxiv.org/abs/2309.07249 |