Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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



Pinned algebraic distances determined by Cartesian products in $ \mathbb{F}_p^2$

Author: Giorgis Petridis
Journal: Proc. Amer. Math. Soc. 145 (2017), 4639-4645
MSC (2010): Primary 11B30
Published electronically: May 26, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ p$ be an odd prime and $ A \subseteq \mathbb{F}_p$ be a subset of the finite field with $ p$ elements. We show that $ A \times A \subseteq \mathbb{F}_p^2$ determines at least a constant multiple of $ \min \{p, \vert A\vert^{3/2}\}$ distinct pinned algebraic distances.

References [Enhancements On Off] (What's this?)

  • [1] E. Aksoy Yazici, B. Murphy, M. Rudnev, and I. D. Shkredov, Growth estimates in positive characteristic via collisions, accepted by Int. Math. Res. Not. IMRN, arXiv:1512.06613, 2015.
  • [2] J. Bourgain, N. Katz, and T. Tao, A sum-product estimate in finite fields, and applications, Geom. Funct. Anal. 14 (2004), no. 1, 27-57. MR 2053599,
  • [3] 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,
  • [4] Fan R. K. Chung, E. Szemerédi, and W. T. Trotter, The number of different distances determined by a set of points in the Euclidean plane, Discrete Comput. Geom. 7 (1992), no. 1, 1-11. MR 1134448,
  • [5] P. Erdös, On sets of distances of $ n$ points, Amer. Math. Monthly 53 (1946), 248-250. MR 0015796,
  • [6] Larry Guth, Polynomial methods in combinatorics, University Lecture Series, vol. 64, American Mathematical Society, Providence, RI, 2016. MR 3495952
  • [7] Larry Guth and Nets Hawk Katz, On the Erdős distinct distances problem in the plane, Ann. of Math. (2) 181 (2015), no. 1, 155-190. MR 3272924,
  • [8] Brandon Hanson, Ben Lund, and Oliver Roche-Newton, On distinct perpendicular bisectors and pinned distances in finite fields, Finite Fields Appl. 37 (2016), 240-264. MR 3426588,
  • [9] 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,
  • [10] Harald Andrés Helfgott and Misha Rudnev, An explicit incidence theorem in $ \mathbb{F}_p$, Mathematika 57 (2011), no. 1, 135-145. MR 2764161,
  • [11] A. Iosevich and M. Rudnev, Erdős distance problem in vector spaces over finite fields, Trans. Amer. Math. Soc. 359 (2007), no. 12, 6127-6142. MR 2336319,
  • [12] Timothy G. F. Jones, An improved incidence bound for fields of prime order, European J. Combin. 52 (2016), part A, 136-145. MR 3425971,
  • [13] N. H. Katz, The flecnode polynomial: A central object in incidence geometry, arxiv:1404.3412, 2014.
  • [14] Nets Hawk Katz and Gábor Tardos, A new entropy inequality for the Erdős distance problem, Towards a theory of geometric graphs, Contemp. Math., vol. 342, Amer. Math. Soc., Providence, RI, 2004, pp. 119-126. MR 2065258,
  • [15] Brendan Murphy and Giorgis Petridis, A point-line incidence identity in finite fields, and applications, Mosc. J. Comb. Number Theory 6 (2016), no. 1, 64-95. MR 3529321
  • [16] G. Petridis, Products of differences in prime order finite fields, arXiv:1602.02142, 2016.
  • [17] Helmut Pottmann and Johannes Wallner, Computational line geometry, Mathematics and Visualization, Springer-Verlag, Berlin, 2001. MR 1849803
  • [18] Oliver Roche-Newton, Misha Rudnev, and Ilya D. Shkredov, New sum-product type estimates over finite fields, Adv. Math. 293 (2016), 589-605. MR 3474329,
  • [19] M. Rudnev, On the number of incidences between planes and points in three dimensions, Combinatorica, to appear.
  • [20] Misha Rudnev and J. M. Selig, On the use of the Klein quadric for geometric incidence problems in two dimensions, SIAM J. Discrete Math. 30 (2016), no. 2, 934-954. MR 3499552,
  • [21] M. Rudnev, I. D. Shkredov, and S. Stevens, On the energy variant of the sum-product conjecture, preprint, arXiv:1607.05053.
  • [22] S. Stevens and F. de Zeeuw, An improved point-line incidence bound over arbitrary fields, arXiv:1609.06284, 2016.
  • [23] Endre Szemerédi and William T. Trotter Jr., Extremal problems in discrete geometry, Combinatorica 3 (1983), no. 3-4, 381-392. MR 729791,

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11B30

Retrieve articles in all journals with MSC (2010): 11B30

Additional Information

Giorgis Petridis
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602

Received by editor(s): October 12, 2016
Received by editor(s) in revised form: December 1, 2016
Published electronically: May 26, 2017
Additional Notes: The author was supported by NSF DMS Grant 1500984
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society