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Simplices over finite fields


Author: Hans Parshall
Journal: Proc. Amer. Math. Soc. 145 (2017), 2323-2334
MSC (2010): Primary 11B30, 05D10, 11T24
DOI: https://doi.org/10.1090/proc/13493
Published electronically: January 25, 2017
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Abstract: We prove that, provided $ d > k$, every sufficiently large subset of $ \mathbf {F}_q^d$ contains an isometric copy of every $ k$-simplex that avoids spanning a nontrivial self-orthogonal subspace. We obtain comparable results for simplices exhibiting self-orthogonal behavior.


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  • [1] J. Bourgain, A Szemerédi type theorem for sets of positive density in $ {\bf R}^k$, Israel J. Math. 54 (1986), no. 3, 307-316. MR 853455, https://doi.org/10.1007/BF02764959
  • [2] Jeremy Chapman, M. Burak Erdoğan, Derrick Hart, Alex Iosevich, and Doowon Koh, Pinned distance sets, $ k$-simplices, Wolff's exponent in finite fields and sum-product estimates, Math. Z. 271 (2012), no. 1-2, 63-93. MR 2917133, https://doi.org/10.1007/s00209-011-0852-4
  • [3] Hillel Furstenberg, Yitzchak Katznelson, and Benjamin Weiss, Ergodic theory and configurations in sets of positive density, Mathematics of Ramsey theory, Algorithms Combin., vol. 5, Springer, Berlin, 1990, pp. 184-198. MR 1083601, https://doi.org/10.1007/978-3-642-72905-8_13
  • [4] Derrick Hart and Alex Iosevich, Ubiquity of simplices in subsets of vector spaces over finite fields, Anal. Math. 34 (2008), no. 1, 29-38 (English, with English and Russian summaries). MR 2379694, https://doi.org/10.1007/s10476-008-0103-z
  • [5] A. Iosevich and M. Rudnev, Erdős distance problem in vector spaces over finite fields, Trans. Amer. Math. Soc. 359 (2007), no. 12, 6127-6142 (electronic). MR 2336319, https://doi.org/10.1090/S0002-9947-07-04265-1
  • [6] Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. MR 2061214
  • [7] Rudolf Lidl and Harald Niederreiter, Finite fields, With a foreword by P. M. Cohn, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 20, Cambridge University Press, Cambridge, 1997. MR 1429394
  • [8] Neil Lyall and Ákos Magyar, Product of simplices and sets of positive upper density in $ {\bf R}^d$, preprint, arXiv:1605.04890 (2016).
  • [9] Ákos Magyar, $ k$-point configurations in sets of positive density of $ \mathbb{Z}^n$, Duke Math. J. 146 (2009), no. 1, 1-34. MR 2475398, https://doi.org/10.1215/00127094-2008-060
  • [10] Le Anh Vinh, On the solvability of systems of bilinear equation in finite fields, Proc. Amer. Math. Soc. 137 (2009), no. 9, 2889-2898. MR 2506446, https://doi.org/10.1090/S0002-9939-09-09947-X

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Additional Information

Hans Parshall
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602

DOI: https://doi.org/10.1090/proc/13493
Received by editor(s): July 20, 2016
Published electronically: January 25, 2017
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2017 American Mathematical Society

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