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Hart, Iosevich, Koh, and Rudnev [Averages over Hyperplanes, Sum-Product Theory in Vector Spaces over Finite Fields and the Erdos–Falconer Distance Conjecture, arXiv:0707.3473v2, 2007] show, using Fourier analysis method, that the finite Erdos–Falconer distance conjecture holds for subsets of the unit sphere in qd. In this article, we give a graph theoretic proof of this result.

The Erdös–Falconer Distance Problem on the Unit Sphere in Vector Spaces Over Finite Fields