Abstract
Using Fourier analysis, Covert, Hart, Iosevich, and Uriarte-Tuero (2008) showed that if the cardinality of a subset of the 2-dimensional vector space over a finite field with q elements is ≥ ρq 2, with \({q^{-1/2} \ll \rho \leq 1}\) then it contains an isometric copy of ≥ cρq 3 triangles. In this note, we give a graph theoretic proof of this result.
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Vinh, L.A. On a Furstenberg-Katznelson-Weiss Type Theorem over Finite Fields. Ann. Comb. 15, 541–547 (2011). https://doi.org/10.1007/s00026-011-0107-4
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DOI: https://doi.org/10.1007/s00026-011-0107-4