On equidistributed directions in finite affine planes

Abstract

We investigate a recently introduced generalisation of determined directions in affine planes. A direction $(d)$ in an affine plane is a point of the line at infinity in the projective completion, and hence corresponds to a parallel class. A (multi)set $S$ of points is equidistributed from direction $(d)$ if all lines from that parallel class intersect $S$ in the same number of points. In this paper, we give a construction of point (multi)sets that are inequidistributed from exactly 3 directions in any finite translation plane, and prove that all such (multi)sets arise from this construction, generalising a result of Kiss and Somlai in Desarguesian planes of prime order. We also use ideas from algebraic graph theory to prove results on directions associated with a pair of point sets $S$ and $T$. If for every direction $(d)$ either $S$ or $T$ is equidistributed from $(d)$, we prove that $|S \cap T| = |S||T|/q^2$, where $q$ is the order of the affine plane. In the appendix, we give a combinatorial alternative approach to proving this equality, which can be of independent interest. We also introduce the notion of directions cross-determined by a pair of sets $S$ and $T$, and prove that $|S| |T| \leq q^2$ if $(S,T)$ cross-determines at most half of the directions, where equality forces $S = T$.