Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

IMPACT FACTOR 2018: 0.867

CiteScore 2018: 0.71

SCImago Journal Rank (SJR) 2018: 0.898
Source Normalized Impact per Paper (SNIP) 2018: 0.964

Mathematical Citation Quotient (MCQ) 2018: 0.71

See all formats and pricing
More options …
Volume 29, Issue 2


Incidences between points and generalized spheres over finite fields and related problems

Nguyen D. Phuong / Pham Thang / Le A. Vinh
Published Online: 2016-06-02 | DOI: https://doi.org/10.1515/forum-2015-0024


Let 𝔽q be a finite field of q elements, where q is a large odd prime power and


where 2ciN, gcd(ci,q)=1, and ai𝔽q for all 1id. A Q -sphere is a set of the form


where 𝒃𝔽qd,r𝔽q. We prove bounds on the number of incidences between a point set 𝒫 and a Q-sphere set 𝒮, denoted by I(𝒫,𝒮), as the following:


We also give a version of this estimate over finite cyclic rings /q, where q is an odd integer. As a consequence of the above bounds, we give an estimate for the pinned distance problem and a bound on the number of incidences between a random point set and a random Q-sphere set in 𝔽qd. We also study the finite field analogues of some combinatorial geometry problems, namely, the number of generalized isosceles triangles, and the existence of a large subset without repeated generalized distances.

Keywords: Point-circle incidences; sum-product estimate; finite fields

MSC 2010: 52C10


  • [1]

    Agarwal P. K. and Pach J., Combinatorial Geometry, John Wiley, New York, 1995. Google Scholar

  • [2]

    Alon N. and Spencer J. H., The Probabilistic Method, 2nd ed., Wiley-Interscience, Hoboken, 2000. Google Scholar

  • [3]

    Bennett M., Iosevich A. and Pakianathan J., Three-point configurations determined by subsets of 𝔽q2 via the Elekes–Sharir paradigm, Combinatorica 34 (2014), no. 6, 689–706. Google Scholar

  • [4]

    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

  • [5]

    Chapman J., Erdoǧan M., Hart D., Iosevich A. and Koh D., Pinned distance sets, k-simplices, Wolff’s exponent in finite fields and sum-product estimates, Math. Z. 271 (2012), no. 1–2, 63–93. Google Scholar

  • [6]

    Charalambides M., A note on distinct distance subsets, J. Geom. 104 (2013), 439–442. Google Scholar

  • [7]

    Cilleruelo J., Combinatorial problems in finite fields and Sidon sets, Combinatorica 32 (2012), no. 5, 497–511. Google Scholar

  • [8]

    Cilleruelo J., Iosevich A., Lund B., Roche-Newton O. and Rudnev M., Elementary methods for incidence problems in finite fields, preprint 2014, http://arxiv.org/abs/1407.2397.

  • [9]

    Conlon D., Fox J., Gasarch W., Harris D. G., Ulrich D. and Zbarsky S., Distinct volume subsets, SIAM J. Discrete Math. 29 (2015), no. 1, 472–480. Google Scholar

  • [10]

    Koh D. and Shen C.-Y., The generalized Erdős–Falconer distance problems in vector spaces over finite fields, J. Number Theory 132 (2012), no. 11, 2455–2473. Google Scholar

  • [11]

    Lefmann H. and Thiele T., Point sets with distinct distances, Combinatorica 15 (1995), 379–408. Google Scholar

  • [12]

    Nguyen H. H., On two-point configurations in a random set, Integers 9 (2009), 41–45. Google Scholar

  • [13]

    Pach J. and Tardos G., Isosceles triangles determined by a planar point set, Graphs Combin. 18 (2002), 769–779. Google Scholar

  • [14]

    Solymosi J., Incidences and the spectra of graphs, Building Bridges Between Mathematics and Computer Science, Bolyai Soc. Math. Stud. 19, Springer, Berlin (2008), 499–513. Google Scholar

  • [15]

    Spencer J., Turán’s theorem for k-graphs, Discrete Math. 2 (1972), 183–186. Google Scholar

  • [16]

    Thang P. V. and Vinh L. A., Erdős–Rényi graph, Szemerédi–Trotter type theorem, and sum-product estimates over finite rings, Forum Math. 27 (2015), no. 1, 331–342. Google Scholar

  • [17]

    Vinh L. A., A Szemerédi–Trotter type theorem and sum-product estimate over finite fields, European J. Combin. 32 (2011), no. 8, 1177–1181. Google Scholar

  • [18]

    Vinh L. A., On a Furstenberg–Katznelson–Weiss type theorem over finite fields, Ann. Comb. 15 (2011), 541–547. Google Scholar

  • [19]

    Vinh L. A., On the generalized Erdős–Falconer distance problems over finite fields, J. Number Theory 133 (2013), 2939–2947. Google Scholar

  • [20]

    Vinh L. A., On point-line incidences in vector spaces over finite fields, Discrete Appl. Math. 177 (2014), 146–151. Google Scholar

  • [21]

    Vinh L. A., The solvability of norm, bilinear and quadratic equations over finite fields via spectra of graph, Forum Math. 26 (2014), no. 1, 141–175. Google Scholar

  • [22]

    Vinh L. A., Product graphs, sum-product graphs and sum-product estimate over finite rings, Forum Math. 27 (2015), no. 3, 1639–1655. Google Scholar

  • [23]

    Vu V., Sum-product estimates via directed expanders, Math. Res. Lett. 15 (2008), 375–388. Google Scholar

About the article

Received: 2015-02-05

Revised: 2016-01-26

Published Online: 2016-06-02

Published in Print: 2017-03-01

The first and third author were supported by Vietnam National Foundation for Science and Technology grant 101.99-2013.21. The work of the second author was partially supported by Swiss National Science Foundation grants 200020-144531 and 200021-137574.

Citation Information: Forum Mathematicum, Volume 29, Issue 2, Pages 449–456, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2015-0024.

Export Citation

© 2017 by De Gruyter.Get Permission

Comments (0)

Please log in or register to comment.
Log in