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 (Rédei, Lückenhafte Polynome über endlichen Körpern, 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<2 p/3.
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