  In this paper we show that if one has a grid A x B, where A and B are sets of
n real numbers, then there can be only very few ``rich'' lines in certain quite
small families. Indeed, we show that if the family has lines taking on
n^epsilon distinct slopes, and where each line is parallel to n^epsilon others
(so, at least n^(2 epsilon) lines in total), then at least one of these lines
must fail to be ``rich''. This result immediately implies non-trivial
sum-product inequalities; though, our proof makes use of the Szemeredi-Trotter
inequality, which Elekes used in his argument for lower bounds on |C+C| +
|C.C|.
