The recent breakthrough by Guth and Katz on the Erdos distinct distance problem in the plane has generated great interest in related distance questions. One line of inquiry concerns the structure of point sets that determine very few distinct distances. In this direction, Erdos conjectured that point sets determining few distinct distances must have a "lattice" structure. While this conjecture is still far out of reach, I will discuss some recent results on the structure of these sets.
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