Möbius disjointness conjecture for Furstenberg's flow on $\mathbb{T}^ω$ in short intervals
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917417871998976 |
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| author | He, Shuyang Liu, Qingyang Ma, Jing |
| author_facet | He, Shuyang Liu, Qingyang Ma, Jing |
| contents | Furstenberg's flow on the infinite-dimensional torus $\mathbb{T}^ω$ is defined by
\[
T (x_1, x_2, \ldots, x_ν, \ldots)
= (x_1 + α, x_2 + h(x_1), \ldots, x_ν+ h(x_1 + (ν-2)β), \ldots)
\] with $α\in \mathbb{R}$ satisfying certain diophantine conditions, $β\in \mathbb{R}\backslash\mathbb{Q},$ and $h: \mathbb{R}\to \mathbb{R}$ being $1$-periodic and analytic. This flow is irregular in the sense that its Birkhoff average does not exist for some $x\in \mathbb{T}^ω$, and it is a generalization of Furstenberg's irregular flow on $\mathbb{T}^2$. The main result of this paper is that the Möbius Disjointness Conjecture of Sarnak holds for the above flow $(\mathbb{T}^ω, T)$ in short intervals $(N-M, N]$ with $N^{5/8+\varepsilon} \leqslant M\leqslant N$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_16840 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Möbius disjointness conjecture for Furstenberg's flow on $\mathbb{T}^ω$ in short intervals He, Shuyang Liu, Qingyang Ma, Jing Number Theory Furstenberg's flow on the infinite-dimensional torus $\mathbb{T}^ω$ is defined by \[ T (x_1, x_2, \ldots, x_ν, \ldots) = (x_1 + α, x_2 + h(x_1), \ldots, x_ν+ h(x_1 + (ν-2)β), \ldots) \] with $α\in \mathbb{R}$ satisfying certain diophantine conditions, $β\in \mathbb{R}\backslash\mathbb{Q},$ and $h: \mathbb{R}\to \mathbb{R}$ being $1$-periodic and analytic. This flow is irregular in the sense that its Birkhoff average does not exist for some $x\in \mathbb{T}^ω$, and it is a generalization of Furstenberg's irregular flow on $\mathbb{T}^2$. The main result of this paper is that the Möbius Disjointness Conjecture of Sarnak holds for the above flow $(\mathbb{T}^ω, T)$ in short intervals $(N-M, N]$ with $N^{5/8+\varepsilon} \leqslant M\leqslant N$. |
| title | Möbius disjointness conjecture for Furstenberg's flow on $\mathbb{T}^ω$ in short intervals |
| topic | Number Theory |
| url | https://arxiv.org/abs/2604.16840 |