Together with Janos Pach I proved that n points on a constant-degree algebraic curve in the real plane determine at least cn^{4/3} distinct distances, unless that curve contains a line or a circle, in which case the number of distances can be linear. The proof is based on a setup recently introduced by Sharir, Sheffer, and Solymosi, which defines a new set of algebraic curves from the point set, and then applies an incidence bound to these curves. I will introduce the background and context of this problem, and show as much of the proof as possible.
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