Special directions on the finite affine plane (Q6591962)

Let \(S\) be a set of points in the affine plane over the finite field of \(p\) elements, where \(p\) is a prime. The set of directions of \(S\) consists of the set of intersection points between the lines connecting two points of \(S\) and the line at infinity. We say that \(S\) is equidistributed in a direction if it intersects the lines with the corresponding fixed slope in the same amount of points. We call a direction special if \(S\) is not equidistributed in that direction.\N\NIn [Discrete Math. 343, No. 5, Article ID 111811, 12 p. (2020; Zbl 1436.51009)], \textit{L. Ghidelli} proved that a set of cardinality \(np\) \((1\leq n\leq p,\ n\in\mathbb{Z})\), is either a set of parallel lines or it has at least \(\lceil\frac{p+n+2}{p+1}\rceil\) special directions. Moreover, it was asked by Ghidelli [loc. cit.] whether the sets that are not the union of a set of parallel lines determine at least \(\frac{p+3}{2}\) special directions.\N\NIn the present paper, the authors show that there is a unique set \(S\) of size \(\frac{p(p-1)}{2}\) in \(\mathbb{F}_{p}^{2}\) which is equidistributed in \(p-2\) directions. Moreover, sets having exactly \(3\) special directions are characterized up to an affine transformation in \(\mathrm{AGL}(2,p)\). This gives a negative answer to Ghidelli's question [loc. cit.], and it shows that this bound is tight for \(n=\frac{p-1}{2}\).