Lattice points inside a random shifted integer polygon
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917721549045760 |
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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 $[0,1]^d$. Consider the value $X_C$ equal to the number of lattice points $\mathbb Z^d$ inside the body $C$ shifted by $X$. It is well known that $\mathbb E X_C = \mathrm{vol}(C)$. The question arises: what can be said about the variance of this random variable? This paper answers this question in the case when $C$ is a polygon with vertices at integer points. Moreover, an explicit distribution of $X_T$ is given for the integer triangle $T$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_09878 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lattice points inside a random shifted integer polygon Tokmachev, Aleksandr Probability Metric Geometry Consider a convex body $C \subset \mathbb{R}^d$. Let $X$ be a random point with uniform distribution in $[0,1]^d$. Consider the value $X_C$ equal to the number of lattice points $\mathbb Z^d$ inside the body $C$ shifted by $X$. It is well known that $\mathbb E X_C = \mathrm{vol}(C)$. The question arises: what can be said about the variance of this random variable? This paper answers this question in the case when $C$ is a polygon with vertices at integer points. Moreover, an explicit distribution of $X_T$ is given for the integer triangle $T$. |
| title | Lattice points inside a random shifted integer polygon |
| topic | Probability Metric Geometry |
| url | https://arxiv.org/abs/2407.09878 |