  We study the structure of planar point sets that determine a small number of
distinct distances. Specifically, we show that if a set P of n points
determines o(n) distinct distances, then no line contains \Omega(n^{7/8})
points of P and no circle contains \Omega(n^{5/6}) points of P.
  We rely on the bipartite and partial variant of the Elekes-Sharir framework
that was presented by Sharir, Sheffer, and Solymosi in \cite{SSS13}. For the
case of lines we combine this framework with a theorem from additive
combinatorics, and for the case of circles we combine it with some basic
algebraic geometry and a recent incidence bound for plane algebraic curves by
Wang, Yang, and Zhang \cite{WYZ13}. A significant difference between our
approach and that of \cite{SSS13} (and other recent extensions) is that,
instead of dealing with distances between two point sets that are restricted to
one-dimensional curves, we consider distances between one set that is
restricted to a curve and one set with no restrictions on it.
