Functions determining $(q+3)/2$ directions over the finite field with $q$ elements

Abstract

Let $q=p^n$, where $p>2$ is prime. We prove that every $q$-set $U\subseteq \mathrm{AG}(2,q)$ determining exactly $(q+3)/2$ directions is affinely equivalent to the graph of $x^{(q+1)/2}$. For $n\ge 2$, we also prove that a $q$-set determining at most $(q+p)/2$ directions is either an affine ${\mathbb{F}}_p$-linear set or affinely equivalent to this graph. The classification for $n=1$ is due to Lovász and Schrijver. For $n=2$, the classification and the stated stability result follow from results of Gács, Lovász, and Sz\H onyi and the direction theorems of Blokhuis, Ball, Brouwer, Storme, and Sz\H onyi, completed by Ball. These direction theorems establish affine ${\mathbb{F}}_p$-linearity for functions with fewer than $(q+3)/2$ determined directions. Our proof uses Rédei polynomials, results of Gács, Lovász, and Sz\H onyi on their coefficients, and an earlier polynomial comparison lemma of the author.