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 R\'edei polynomial with Sz\H{o}nyi'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.