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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.11087 |
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| _version_ | 1866912962220916736 |
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| author | Liu, Qingyang Ma, Jing Wang, Hongbo |
| author_facet | Liu, Qingyang Ma, Jing Wang, Hongbo |
| contents | Let $\mathbb{T}^ω$ be the infinite-dimensional torus, and $T: \mathbb{T}^ω\to \mathbb{T}^ω$ be defined by \[
T: (x_1, x_2, \dots, x_k, \ldots) \mapsto (x_1 + α, x_2 + h(x_1), \dots, x_k + h(x_1 + (k-2)β), \dots) \] with $α\in \mathbb{R}, β\in \mathbb{R}\backslash\mathbb{Q},$ and $h: \mathbb{R}\to \mathbb{R}$ being $1$-period and $C^{1+\varepsilon}$-smooth. This flow $(\mathbb{T}^ω, T)$ is distal, and is also irregular in the sense that its Birkhoff average does not exist for all $x\in \mathbb{T}^ω$. The main result of this paper is that the M öbius Disjointness Conjecture of Sarnak holds for $(\mathbb{T}^ω, T)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_11087 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The M öbius Disjointness Conjecture on infinite-dimensional torus Liu, Qingyang Ma, Jing Wang, Hongbo Number Theory Dynamical Systems Let $\mathbb{T}^ω$ be the infinite-dimensional torus, and $T: \mathbb{T}^ω\to \mathbb{T}^ω$ be defined by \[ T: (x_1, x_2, \dots, x_k, \ldots) \mapsto (x_1 + α, x_2 + h(x_1), \dots, x_k + h(x_1 + (k-2)β), \dots) \] with $α\in \mathbb{R}, β\in \mathbb{R}\backslash\mathbb{Q},$ and $h: \mathbb{R}\to \mathbb{R}$ being $1$-period and $C^{1+\varepsilon}$-smooth. This flow $(\mathbb{T}^ω, T)$ is distal, and is also irregular in the sense that its Birkhoff average does not exist for all $x\in \mathbb{T}^ω$. The main result of this paper is that the M öbius Disjointness Conjecture of Sarnak holds for $(\mathbb{T}^ω, T)$. |
| title | The M öbius Disjointness Conjecture on infinite-dimensional torus |
| topic | Number Theory Dynamical Systems |
| url | https://arxiv.org/abs/2603.11087 |