Hostname: page-component-5cf477f64f-r2nwp Total loading time: 0 Render date: 2025-04-05T07:38:23.463Z Has data issue: false hasContentIssue false
  • English
  • Français

ON POINT SETS IN VECTOR SPACES OVER FINITE FIELDS THAT DETERMINE ONLY ACUTE ANGLE TRIANGLES

Part of: Discrete geometry Designs and configurations

Published online by Cambridge University Press:  21 October 2009

IGOR E. SHPARLINSKI
IGOR E. SHPARLINSKI*
Affiliation:
Department of Computing, Macquarie University, Sydney, NSW 2109, Australia (email: igor@ics.mq.edu.au)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For three points , and in the n-dimensional space 𝔽nq over the finite field 𝔽q of q elements we give a natural interpretation of an acute angle triangle defined by these points. We obtain an upper bound on the size of a set 𝒵 such that all triples of distinct points define acute angle triangles. A similar question in the real space ℛn dates back to P. Erdős and has been studied by several authors.

Keywords

acute angle trianglefinite fieldcharacter sum

MSC classification

Secondary: 05B25: Finite geometries 52C10: Erdős problems and related topics of discrete geometry
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 81 , Issue 1 , February 2010 , pp. 114 - 120
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

References

[1]Ackerman, E. and Ben-Zwi, O., ‘On sets of points that determine only acute angles’, European J. Combin. 30 (2009), 908–910.Google Scholar
[2]Bourgain, J. and Garaev, M. Z., ‘On a variant of sum–product estimates and explicit exponential sum bounds in prime fields’, Math. Proc. Cambridge Philos. Soc. 146 (2008), 1–21.CrossRefGoogle Scholar
[3]Bourgain, J., Katz, N. and Tao, T., ‘A sum product estimate in finite fields and applications’, Geom. Funct. Anal. 14 (2004), 27–57.Google Scholar
[4]Covert, D., Hart, D., Iosevich, A., Koh, D. and Rudnev, M., ‘Generalized incidence theorems, homogeneous forms and sum–product estimates in finite fields’, European J. Combin., to appear.Google Scholar
[5]Dvir, Z., ‘On the size of Kakeya sets in finite fields’, J. Amer. Math. Soc. 22 (2009), 1093–1097.CrossRefGoogle Scholar
[6]Garaev, M. Z., ‘The sum–product estimate for large subsets of prime fields’, Proc. Amer. Math. Soc. 136 (2008), 2735–2739.Google Scholar
[7]Graham, S. W. and Ringrose, C. J., ‘Lower bounds for least quadratic nonresidues’, in: Analytic Number Theory, Allerton Park 1989, Progress in Mathematics, 85 (Birkhäuser, Basel, 1990), pp. 269–309.Google Scholar
[8]Hart, D. and Iosevich, A., ‘Ubiquity of simplices in subsets of vector spaces over finite fields’, Anal. Math. 34 (2007), 29–38.CrossRefGoogle Scholar
[9]Hart, D., Iosevich, A., Koh, D. and Rudnev, M., ‘Averages over hyperplanes, sum–product theory in finite fields’, Trans. Amer. Math. Soc., to appear.Google Scholar
[10]Hart, D., Iosevich, A. and Solymosi, J., ‘Sums and products in finite fields via Kloosterman sums’, Int. Math. Res. Not. 2007 (2007), 1–14; Article ID rnm007.Google Scholar
[11]Iosevich, A. and Koh, D., ‘Erdős–Falconer distance problem, exponential sums, and Fourier analytic approach to incidence theorems in vector spaces over finite fields’, SIAM J. Discrete Math. 23 (2008), 123–135.Google Scholar
[12]Iosevich, A. and Rudnev, M., ‘Erdős distance problem in vector spaces over finite fields’, Trans. Amer. Math. Soc. 359 (2007), 6127–6142.CrossRefGoogle Scholar
[13]Iosevich, A. and Senger, S., ‘Orthogonal systems in vector spaces over finite fields’, Electron. J. Combin. 15 (2008), Article R151.CrossRefGoogle Scholar
[14]Iosevich, A., Shparlinski, I. E. and Xiong, M., ‘Sets with integral distances in finite fields’, Trans. Amer. Math. Soc., to appear.Google Scholar
[15]Iwaniec, H. and Kowalski, E., Analytic Number Theory (American Mathematical Society, Providence, RI, 2004).Google Scholar
[16]Katz, N. M., ‘Estimates for nonsingular multiplicative character sums’, Int. Math. Res. Not. 2002 (2002), 333–349.CrossRefGoogle Scholar
[17]Katz, N. H. and Shen, C.-Y., ‘Garaev’s inequality in finite fields not of prime order’, J. Anal. Combin. 3 (2008), Article 3.Google Scholar
[18]Perel’muter, G. I., ‘Estimation of a multiple sum with the Legendre symbol’, Mat. Zametki 18 (1975), 421–427 (in Russian).Google Scholar
[19]Shen, C.-Y., ‘An extension of Bourgain and Garaev’s sum–product estimates’, Acta Arith. 135 (2008), 351–356.Google Scholar
[20]Vinh, L. A., ‘Explicit Ramsey graphs and Erdős distance problem over finite Euclidean and non-Euclidean spaces’, Electron. J. Combin. 15 (2008), Aricle R5.Google Scholar
[21]Vinh, L. A., ‘On the number of orthogonal systems in vector spaces over finite fields’, Electron. J. Combin. 15 (2008), Aricle N32.Google Scholar
[22]Vinh, L. A., ‘On a Furstenberg–Katznelson–Weiss type theorem over finite fields’, Ann. Combin., to appear.Google Scholar
[23]Vinh, L. A., ‘Szemeredi–Trotter type theorem and sum–product estimate in finite fields’, European J. Combin., to appear.Google Scholar
[24]Wildberger, N. J., Divine Proportions: Rational Trigonometry to Universal Geometry (Wild Egg Books, Sydney, 2005).Google Scholar