Geodesic
Distinct triangle areas in a planar point set over finite fields
The electronic journal of combinatorics, Tome 18 (2011) no. 1
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Let $\mathcal{P}$ be a set of $n$ points in the finite plane $\mathbb{F}_q^2$ over the finite field $\mathbb{F}_q$ of $q$ elements, where $q$ is an odd prime power. For any $s \in \mathbb{F}_q$, denote by $A (\mathcal{P}; s)$ the number of ordered triangles whose vertices in $\mathcal{P}$ having area $s$. We show that if the cardinality of $\mathcal{P}$ is large enough then $A (\mathcal{P}; s)$ is close to the expected number $|\mathcal{P}|^3/q$.
DOI : 10.37236/700
Classification : 05C35, 05C38 
@article{10_37236_700,
     author = {Le Anh Vinh},
     title = {Distinct triangle areas in a planar point set over finite fields},
     journal = {The electronic journal of combinatorics},
     year = {2011},
     volume = {18},
     number = {1},
     doi = {10.37236/700},
     zbl = {1229.05158},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/700/}
}
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Le Anh Vinh. Distinct triangle areas in a planar point set over finite fields. The electronic journal of combinatorics, Tome 18 (2011) no. 1. doi: 10.37236/700

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