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

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## 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 $\min \{p, |A|^{3/2}\}$ distinct pinned algebraic distances.## References

- 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. - 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**, DOI 10.1007/s00039-004-0451-1 - 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**, DOI 10.1007/s00209-011-0852-4 - 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**, DOI 10.1007/BF02187820 - P. Erdös,
*On sets of distances of $n$ points*, Amer. Math. Monthly**53**(1946), 248–250. MR**15796**, DOI 10.2307/2305092 - Larry Guth,
*Polynomial methods in combinatorics*, University Lecture Series, vol. 64, American Mathematical Society, Providence, RI, 2016. MR**3495952**, DOI 10.1090/ulect/064 - 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**, DOI 10.4007/annals.2015.181.1.2 - 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**, DOI 10.1016/j.ffa.2015.10.002 - 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 - Harald Andrés Helfgott and Misha Rudnev,
*An explicit incidence theorem in $\Bbb F_p$*, Mathematika**57**(2011), no. 1, 135–145. MR**2764161**, DOI 10.1112/S0025579310001208 - 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**, DOI 10.1090/S0002-9947-07-04265-1 - Timothy G. F. Jones,
*An improved incidence bound for fields of prime order*. part A, European J. Combin.**52**(2016), no. part A, 136–145. MR**3425971**, DOI 10.1016/j.ejc.2015.09.004 - N. H. Katz,
*The flecnode polynomial: A central object in incidence geometry*, arxiv:1404.3412, 2014. - 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**, DOI 10.1090/conm/342/06136 - 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** - G. Petridis,
*Products of differences in prime order finite fields*, arXiv:1602.02142, 2016. - Helmut Pottmann and Johannes Wallner,
*Computational line geometry*, Mathematics and Visualization, Springer-Verlag, Berlin, 2001. MR**1849803** - 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**, DOI 10.1016/j.aim.2016.02.019 - M. Rudnev,
*On the number of incidences between planes and points in three dimensions*, Combinatorica, to appear. - 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**, DOI 10.1137/16M1059412 - M. Rudnev, I. D. Shkredov, and S. Stevens,
*On the energy variant of the sum-product conjecture*, preprint, arXiv:1607.05053. - S. Stevens and F. de Zeeuw,
*An improved point-line incidence bound over arbitrary fields*, arXiv:1609.06284, 2016. - Endre Szemerédi and William T. Trotter Jr.,
*Extremal problems in discrete geometry*, Combinatorica**3**(1983), no. 3-4, 381–392. MR**729791**, DOI 10.1007/BF02579194

## Additional Information

**Giorgis Petridis**- Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
- MR Author ID: 956366
- Email: giorgis@cantab.net
- 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
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**145**(2017), 4639-4645 - MSC (2010): Primary 11B30
- DOI: https://doi.org/10.1090/proc/13649
- MathSciNet review: 3691983