LONG PATHS IN THE DISTANCE GRAPH OVER LARGE SUBSETS OF VECTOR SPACES OVER FINITE FIELDS BENNETT, MICHAEL; CHAPMAN, JEREMY; COVERT, DAVID; HART, DERRICK; IOSEVICH, ALEX; PAKIANATHAN, JONATHAN;
Let , the d-dimensional vector space over the finite field with q elements. Construct a graph, called the distance graph of E, by letting the vertices be the elements of E and connect a pair of vertices corresponding to vectors x, y 2 E by an edge if . We shall prove that the non-overlapping chains of length k, with k in an appropriate range, are uniformly distributed in the sense that the number of these chains equals the statistically correct number, plus a much smaller remainder.
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