On the volume set of point sets in vector spaces over finite fields
HTML articles powered by AMS MathViewer
- by Le Anh Vinh PDF
- Proc. Amer. Math. Soc. 141 (2013), 3067-3071 Request permission
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 $|\mathcal {E}| \geq (d-1)q^{d - 1}$, 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.References
- Omran Ahmadi and Igor E. Shparlinski, Distribution of matrices with restricted entries over finite fields, Indag. Math. (N.S.) 18 (2007), no. 3, 327–337. MR 2373685, DOI 10.1016/S0019-3577(07)00013-4
- David Covert, Derrick Hart, Alex Iosevich, Doowon Koh, and Misha Rudnev, Generalized incidence theorems, homogeneous forms and sum-product estimates in finite fields, European J. Combin. 31 (2010), no. 1, 306–319. MR 2552610, DOI 10.1016/j.ejc.2008.11.015
- Derrick Hart and Alex Iosevich, Sums and products in finite fields: an integral geometric viewpoint, Radon transforms, geometry, and wavelets, Contemp. Math., vol. 464, Amer. Math. Soc., Providence, RI, 2008, pp. 129–135. MR 2440133, DOI 10.1090/conm/464/09080
- Derrick Hart, Alex Iosevich, Doowon Koh, and Misha Rudnev, Averages over hyperplanes, sum-product theory in vector spaces over finite fields and the Erdős-Falconer distance conjecture, Trans. Amer. Math. Soc. 363 (2011), no. 6, 3255–3275. MR 2775806, DOI 10.1090/S0002-9947-2010-05232-8
- Le Anh Vinh, Distribution of determinant of matrices with restricted entries over finite fields, J. Comb. Number Theory 1 (2009), no. 3, 203–212. MR 2681305
- Le Anh Vinh, Singular matrices with restricted rows in vector spaces over finite fields, Discrete Math. 312 (2012), no. 2, 413–418. MR 2852600, DOI 10.1016/j.disc.2011.10.002
- Le Anh Vinh, On the permanents of matrices with restricted entries over finite fields, SIAM J. Discrete Math. 26 (2012), no. 3, 997–1007. MR 3022119, DOI 10.1137/110835050
Additional Information
- Le Anh Vinh
- Affiliation: University of Education, Vietnam National University, Hanoi, Vietnam
- MR Author ID: 798264
- Email: vinhla@vnu.edu.vn
- 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
- © Copyright 2013 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 141 (2013), 3067-3071
- MSC (2010): Primary 11T99
- DOI: https://doi.org/10.1090/S0002-9939-2013-11630-8
- MathSciNet review: 3068960