On rich lines in grids

Abstract

In this paper we show that if one has a grid A×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 nε distinct slopes, and where each line is parallel to nε others (so, at least n2ε lines in total), then at least one of these lines must fail to be "rich". This result immediately implies non-trivial sumproduct inequalities; though, our proof makes use of the Szemeredi-Trotter inequality, which Elekes used in his argument for lower bounds on {pipe}C+C{pipe}+{pipe}C. C{pipe}. © 2010 Springer Science+Business Media, LLC.