On integer points inside a randomly shifted polyhedron
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Accesso online: | |
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| _version_ | 1866916840556462080 |
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| author | Tokmachev, Aleksandr |
| author_facet | Tokmachev, Aleksandr |
| contents | Consider a convex body $C \subset \mathbb{R}^d$. Let $X$ be a random point with uniform distribution in $[0,1]^d$. Define $X_C$ as the number of lattice points in $\mathbb{Z}^d$ inside the translated body $C + X$. It is well known that $\mathbb{E} X_C = \mathrm{vol}(C)$. A natural question arises: What can be said about the distribution of $X_C$ in general? In this work, we study this question when $C$ is a polyhedron with vertices at integer points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_09355 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On integer points inside a randomly shifted polyhedron Tokmachev, Aleksandr Probability Metric Geometry Consider a convex body $C \subset \mathbb{R}^d$. Let $X$ be a random point with uniform distribution in $[0,1]^d$. Define $X_C$ as the number of lattice points in $\mathbb{Z}^d$ inside the translated body $C + X$. It is well known that $\mathbb{E} X_C = \mathrm{vol}(C)$. A natural question arises: What can be said about the distribution of $X_C$ in general? In this work, we study this question when $C$ is a polyhedron with vertices at integer points. |
| title | On integer points inside a randomly shifted polyhedron |
| topic | Probability Metric Geometry |
| url | https://arxiv.org/abs/2507.09355 |