## Abstract

We prove that if \(E \subset \mathbb{F}_Q^2\), *q* ≡ 3 mod 4, has size greater than \(Cq^{\tfrac{7} {4}}\), then *E* determines a positive proportion of all congruence classes of triangles in \(\mathbb{F}_q^2\).

The approach in this paper is based on the approach to the Erdős distance problem in the plane due to Elekes and Sharir, followed by an incidence bound for points and lines in \(\mathbb{F}_q^3\). We also establish a weak lower bound for a related problem in the sense that any subset *E* of \(\mathbb{F}_q^2\) of size less than *cq*
^{4/3} definitely does not contain a positive proportion of translation classes of triangles in the plane. This result is a special case of a result established for *n*-simplices in \(\mathbb{F}_q^d\). Finally, a necessary and sufficient condition on the lengths of a triangle for it to exist in \(\mathbb{F}^2\) for any field \(\mathbb{F}\) of characteristic not equal to 2 is established as a special case of a result for *d*-simplices in \(\mathbb{F}^d\).

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## References

- [1]
J. Bourgain: A Szemerédi type theorem for sets of positive density,

*Israel J. Math.***54**(1986), no. 3, 307–331. - [2]
J. Bourgain, N. Katz and T. Tao: A sum-product estimate in finite flelds, and applications,

*Geom. Func. Anal.***14**(2004), 27–57. - [3]
P. Brass, W. Moser and J. Pach:

*Research Problems in Discrete Geometry*, Springer (2005). - [4]
J. Chapman, B. Erdogan, D. Hart, A. Iosevich and D. Koh: Pinned distance sets,

*k*-simplices, Wolff’s exponent in finite flelds and sum-product estimates,*Math-ematische Zeitschrift*, (online; paper version to appear), (2011). - [5]
K. B. Chilakamarri: Unit-distance graphs in rational

*n*-spaces,*Discrete Math.***69**(1988), 213–218. - [6]
D. Covert, D. Hart, A. Iosevich, S. Senger and I. Uriarte-Tuero: An analog of the Furstenberg-Katznelson-Weiss theorem on triangles in sets of positive density in finite fleld geometries,

*Discrete Math.***311**(2011), 423–430. - [7]
G. Elekes and M. Sharir: Incidences in three dimensions and distinct distances in the plane,

*Combin. Probab. Comput.***20**(2011), 571–608. - [8]
C. Elsholtz, W. Klotz: Maximal Dimension of Unit Simplices,

*Discrete and Computational Geometry,***34**(2005), 167–177. - [9]
P. Erdős: On sets of distances of

*n*points,*Amer. Math. Monthly***53**(1946), 248–250. - [10]
P. Erdős and G. Purdy: Some extremal problems in geometry,

*J. Combin. Theory Ser.***A 10**(1971), 246–252. - [11]
P. Erdős and G. Purdy: Some extremal problems in geometry III,

*Proceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory and Computing*(Florida Atlantic Univ., Boca Raton, Fla., 1975), 291–308, Congressus Numerantium, No. XIV, Utilitas Math., Winnipeg (1975). - [12]
P. Erdős and G. Purdy: Some extremal problems in geometry IV,

*Proceedings of the Seventh Southeastern Conference on Combinatorics, Graph Theory, and Computing*(Louisiana State Univ., Baton Rouge, La., 1976), 307–322, Congressus Numerantium, No. XVII, Utilitas Math., Winnipeg (1976). - [13]
P. Erdős and G. Purdy: Some extremal problems in geometry V,

*Proceedings of the Eighth Southeastern Conference on Combinatorics, Graph Theory and Computing*(Louisiana State Univ., Baton Rouge, La., 1977), 569–578, Congressus Numerantium, No. XIX, Utilitas Math., Winnipeg (1977). - [14]
P. Erdős and G. Purdy: Some combinatorial problems in the plane,

*J. Combin. Theory Ser. A***25**(1978), 205–210. - [15]
P. Erdős and G. Purdy:

*Extremal problems in combinatorial geometry*, in: Handbook of Combinatorics, 2 vols, 809–874, Elsevier Sci. B. V., Amsterdam (1995). - [16]
K. J. Falconer:

*The geometry of fractal sets*, Cambridge Tracts in Mathematics, 85 Cambridge Univ. Pr., Cambridge (1986). - [17]
H. Furstenberg, Y. Katznelson and B. Weiss:

*Ergodic theory and configurations in sets of positive density*, Mathematics of Ramsey theory, 184–198, Algorithms Combin., 5, Springer, Berlin (1990). - [18]
A. Greeneleaf and A. Iosevich: On three point configurations determined by subsets of the Euclidean plane, the associated bilinear operator and applications to discrete geometry,

*Analysis and PDE*(accepted for publication), (2010). - [19]
L. Guth and N. Katz:

*On the Erdős distinct distance problem in the plane*, (preprint) http://arxiv.org/pdf/1011.4105. - [20]
S. Hofmann and A. Iosevich: Circular averages and Falconer/Erdős distance conjecture in the plane for random metrics,

*Proc. Amer. Math. Soc.***133**(2005), 133–143. - [21]
A. Iosevich, H. Jorati and I. Laba: Geometric incidence theorems via Fourier analysis,

*Trans. Amer. Math. Soc.***361**(2009), 6595–6611. - [22]
A. Iosevich, I. Laba:

*K*-distance sets, Falconer conjecture, and discrete analogues,*Integers***5**(2005), 11. - [23]
A. Iosevich and M. Rudnev:

*Erdős distance problem in vector spaces over finite flelds*, Transactions of the American Mathematical Society, (2007). - [24]
P. Mattila:

*Geometry of sets and measures in Euclidean spaces*, Cambridge Univ. Pr.,**44**(1995). - [25]
D. Robinson:

*A course in the theory of groups*, Second Edition, Graduate Texts in Mathematics, Springer,**80**, (1995). - [26]
J. Spencer, E. Szemerédi and W. T. Trotter, Jr.:

*Unit distances in the Euclidean plane*, Graph theory and combinatorics, 293–303, Academic Press, London, (1984). - [27]
E. Szemerédi and W. T. Trotter, Jr.: Extremal problems in discrete geometry,

*Combinatorica***3**(1983), 381–392. - [28]
L. Székely: A. Crossing numbers and hard Erdős problems in discrete geometry

*Combin. Probab. Comput.***6**(1997), 353–358. - [29]
L. Vinh:

*Szemeredi-Trotter type theorem and sum-product estimate in finite flelds*, (preprint), http://arxiv.org/pdf/0711.4427 (2008). - [30]
T. Ziegler: Nilfactors of ℝ

^{d}actions and configurations in sets of positive upper density in ℝ^{m},*J. Anal. Math.***99**(2006), 249–266.

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The authors were supported by NSF grant DMS-1045404.

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Bennett, M., Iosevich, A. & Pakianathan, J. Three-point configurations determined by subsets of \(\mathbb{F}_q^2\) via the Elekes-Sharir Paradigm.
*Combinatorica* **34, **689–706 (2014). https://doi.org/10.1007/s00493-014-2978-6

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### Mathematics Subject Classification (2000)

- 52C10
- 51E99