On the directions determined by a Cartesian product in an affine Galois plane

Abstract

We prove that the number of directions contained in a set of the form $A\times B\subset AG(2,p)$, where $p$ is prime, is at least $|A||B|-\min \{|A|,|B|\}+2$. Here $A$ and $B$ are subsets of $GF(p)$ each with at least two elements and $|A||B|<p$. This bound is tight for an infinite class of examples. Our main tool is the use of the Rédei polynomial with Szőnyi's extension. As an application of our main result, we obtain an upper bound on the clique number of a Paley graph, matching the current best bound obtained recently by Hanson and Petridis.

8 pages