Using methods from arithmetic combinatorics, we prove the following.

Theorem. For every ǫ > 0, there exists δ > 0, such that the following holds for all pairs

of sets A and B of real numbers with

|A| = |B| = n,

where n > n0(ǫ): Suppose one has a family of lines such that

• There are at least n

ǫ different slopes among them; and,

• every line is parallel to at least n

ǫ others.

Then, at least one of the lines must hit the grid A × B in fewer than n

1−δ points; that is,

not all the lines can be “n

1−δ

-rich”.

If one applies the Szemeredi-Trotter inequality to this problem, one will see that the

bounds it gives are nowhere near to what one would need in order to prove it. Furthermore,

the theorem already implies some weak form of sum-product inequalities, so is at least at

that level of depth.

We will mention some conjectures related to this theorem; and, if time permits, we

will present a simple geometric argument that gives somewhat weaker (thought still nontrivial) results.

This is joint work with my student Evan Borenstein.