Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the volume set of point sets in vector spaces over finite fields

Author: Le Anh Vinh
Journal: Proc. Amer. Math. Soc. 141 (2013), 3067-3071
MSC (2010): Primary 11T99
Published electronically: June 4, 2013
MathSciNet review: 3068960
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Abstract: We show that if $ \mathcal {E}$ is a subset of the $ d$-dimensional vector space over a finite field $ \mathbb{F}_q$ ($ d \geq 3$) of cardinality $ \vert\mathcal {E}\vert \geq (d-1)q^{d - 1}$, then the set of volumes of $ d$-dimensional parallelepipeds determined by $ \mathcal {E}$ covers $ \mathbb{F}_q$. This bound is sharp up to a factor of $ (d-1)$, as taking $ \mathcal {E}$ to be a $ (d - 1)$-hyperplane through the origin shows.

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Le Anh Vinh
Affiliation: University of Education, Vietnam National University, Hanoi, Vietnam

Received by editor(s): October 1, 2011
Received by editor(s) in revised form: December 5, 2011
Published electronically: June 4, 2013
Additional Notes: This research is supported by the Vietnam National Foundation for Science and Technology Development, grant No. 101.01-2011.28
Communicated by: Jim Haglund
Article copyright: © Copyright 2013 American Mathematical Society