On rich lines in grids

TL;DR

This paper investigates the limitations on the number of rich lines in a grid of real numbers, establishing that certain small families of lines with many parallel lines cannot all be rich, leading to new sum-product inequalities.

Contribution

It proves that in a grid, families of lines with many parallel lines cannot all be rich, providing new bounds and implications for sum-product problems.

Findings

Few rich lines in small families within a grid

At least one line in such families must not be rich

Results imply new sum-product inequalities

Abstract

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|.