Simultaneous visibility in the algebraic lattice
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908893520592896 |
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| author | Kumar, Rishi Takeda, Wataru |
| author_facet | Kumar, Rishi Takeda, Wataru |
| contents | Let $K$ be a number field with ring of integers $\mathcal{O}$. Two lattice points ${\bf x, y}\in \mathcal{O}^m$ with $m\geq 2$ are said to be visible from one another if $\gcd((x_i-y_i),\ldots, (x_m-y_m))=\mathcal{O}$, where $(x_i-y_i)$ is the ideal generated by $x_i-y_i$. Let $S\subset \mathcal{O}^m$ be a finite set. For $K=\mathbb{Q}$, the asymptotic density of the set of lattice points, visible from all points of $S$, was studied by several authors. For general number fields $K$, however, the asymptotic density has been studied only in the special case $S=\{(0,\ldots,0)\}$. Our main result establishes the corresponding density formula for a number field $K$ whose ring of integers $\mathcal{O}$ is a principal ideal domain, for all finite sets $S$ with $|S|\geq 2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_16332 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Simultaneous visibility in the algebraic lattice Kumar, Rishi Takeda, Wataru Number Theory Primary 11P11, 11R45, Secondary 11P21, 11R04 Let $K$ be a number field with ring of integers $\mathcal{O}$. Two lattice points ${\bf x, y}\in \mathcal{O}^m$ with $m\geq 2$ are said to be visible from one another if $\gcd((x_i-y_i),\ldots, (x_m-y_m))=\mathcal{O}$, where $(x_i-y_i)$ is the ideal generated by $x_i-y_i$. Let $S\subset \mathcal{O}^m$ be a finite set. For $K=\mathbb{Q}$, the asymptotic density of the set of lattice points, visible from all points of $S$, was studied by several authors. For general number fields $K$, however, the asymptotic density has been studied only in the special case $S=\{(0,\ldots,0)\}$. Our main result establishes the corresponding density formula for a number field $K$ whose ring of integers $\mathcal{O}$ is a principal ideal domain, for all finite sets $S$ with $|S|\geq 2$. |
| title | Simultaneous visibility in the algebraic lattice |
| topic | Number Theory Primary 11P11, 11R45, Secondary 11P21, 11R04 |
| url | https://arxiv.org/abs/2603.16332 |