Directions in an affine Galois plane and the clique number of the Paley graph

We prove that the number of directions determined by a set of the form $A \times B \subset AG(2,p)$, where $p$ is prime, is at least $|A||B| - min \{ |A|,|B| \} + 2$. We are using the polynomial method: the Rédei polynomial with Szőnyi's extension + a simple variant of Stepanov's method. As an application of the result, we obtain an upper bound on the clique number of the Paley graph.

Based on joint work with Daniel Di Benedetto and Ethan White.