Abstract

In this paper we prove that if $q$ is the square of a prime and $U$ is a set of $q$ points determining at least $\frac{q+3}{2}$ directions, then either $U$ is affinely equivalent to the graph of the function $x^{\frac{q+1}{2}}$ or it determines at least $\frac{q+p}{2}+1$ directions. This is sharp, the example is due to Polverino, Szőnyi and Weiner. Our method combines the lacunary polynomial and the double power sum approach.

Keywords

Mathematical Subject Classification 2000

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