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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2511.14954 |
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| _version_ | 1866912717380517888 |
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| author | Chow, Sam Zhou, Qing-Long |
| author_facet | Chow, Sam Zhou, Qing-Long |
| contents | Consider a sequence of integral matrices $\mathcal{A}=(A_n)_{n\in\N}$, and a $d$-tuple function ${\bf r}=(r_1,\ldots,r_d)\colon \N\to (0,\frac{1}{2})$. For a fixed vector ${\bm α},$ we are interested in the set $\mathcal{T}_{\bm α}(\mathcal{A}, {\bf r})$ of vectors ${\bm β}\in[0,1)^{d}$ for which $A_n{\bm α}~~\!\!\!\!\!\pmod{1}$ infinitely often lies in the box centred at ${\bm β}$, with side lengths $2r_i(n)$ in each coordinate direction. Under mild conditions on $\mathcal{A}$ and ${\bf r}$, we prove a metric dichotomy for the size of $\mathcal{T}_{\bm α}(\mathcal{A}, {\bf r}),$ valid for almost every ${\bm α}$ with respect to any fractal measure with a certain polynomial Fourier decay rate. Furthermore, removing all restrictions on ${\bf r}$, we establish a metric dichotomy for Lebesgue almost every ${\bm α}.$ This solves a variant of a conjecture of González Robert, Hussain, Shulga and Ward [Conjecture 1.10, Bull. London Math. Soc. 2025]. Finally, we also establish a Jarník-type theorem for $\mathcal{T}_{\bm α}(\mathcal{A}, {\bf r}).$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_14954 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Twisted Diophantine approximation for matrix transformations of tori Chow, Sam Zhou, Qing-Long Number Theory Consider a sequence of integral matrices $\mathcal{A}=(A_n)_{n\in\N}$, and a $d$-tuple function ${\bf r}=(r_1,\ldots,r_d)\colon \N\to (0,\frac{1}{2})$. For a fixed vector ${\bm α},$ we are interested in the set $\mathcal{T}_{\bm α}(\mathcal{A}, {\bf r})$ of vectors ${\bm β}\in[0,1)^{d}$ for which $A_n{\bm α}~~\!\!\!\!\!\pmod{1}$ infinitely often lies in the box centred at ${\bm β}$, with side lengths $2r_i(n)$ in each coordinate direction. Under mild conditions on $\mathcal{A}$ and ${\bf r}$, we prove a metric dichotomy for the size of $\mathcal{T}_{\bm α}(\mathcal{A}, {\bf r}),$ valid for almost every ${\bm α}$ with respect to any fractal measure with a certain polynomial Fourier decay rate. Furthermore, removing all restrictions on ${\bf r}$, we establish a metric dichotomy for Lebesgue almost every ${\bm α}.$ This solves a variant of a conjecture of González Robert, Hussain, Shulga and Ward [Conjecture 1.10, Bull. London Math. Soc. 2025]. Finally, we also establish a Jarník-type theorem for $\mathcal{T}_{\bm α}(\mathcal{A}, {\bf r}).$ |
| title | Twisted Diophantine approximation for matrix transformations of tori |
| topic | Number Theory |
| url | https://arxiv.org/abs/2511.14954 |