On the number of directions determined by a point set in AG(2, p)

Abstract It has been known for a long time that a p-element point set in AG(2, p ), which is not a line, determines at least ( p +3)/2 directions (Redei, Luckenhafte Polynome uber endlichen Korpern, Birkhauser Verlag, Basel, 1970 (English translation: Lacunary Polynomials over Finite Fields, North-Holland, Amsterdam, 1973)). In this paper we look for sets determining more than ( p +3)/2 directions. We prove that besides two examples no set determines ( p +5)/2 directions, give an infinite series of examples determining 7 p /9 directions approximately and prove results about the graph of monomials. These results suggest a conjecture, namely that no point set can determine N directions with ( p +3)/2 N p /3.