On Distance in Some Finite Planes and Graphs Arising from Those Planes

In this paper, affine and projective graphs are obtained from affine and projective planes of order p r by accepting a line as a path. Some properties of these affine and projective graphs are investigated. Moreover, a definition of distance is given in the affine and projective planes of order p r and, with the help of this distance definition, the point or points having the most advantageous (central) position in the corresponding graphs are determined, with some examples being given. In addition, the concepts of a circle, ellipse, hyperbola, and parabola, which are well known for the Euclidean plane, are carried over to these finite planes. Finally, the roles of finite affine and projective Klingenberg planes in all the results obtained are considered and their equivalences in graph applications are discussed.

Point Distribution and Perfect Directions in 𝔽p2\mathbb{F}_p^2

Let p ≥ 3 be a prime, S \subseteq \mathbb{F}_p^2 a nonempty set, and w:\mathbb{F}_p^2 \to R a function with supp w = S. Applying an uncertainty inequality due to András Bíró and the present author, we show that there are at most {1 \over 2}\left| S \right| directions in \mathbb{F}_p^2 such that for every line l in any of these directions, one has \sum\limits_{z \in l} {w\left( z \right) = {1 \over p}\sum\limits_{z \in \mathbb{F}_p^2} {w\left( z \right),} } except if S itself is a line and w is constant on S (in which case all, but one direction have the property in question). The bound {1 \over 2}\left| S \right| is sharp.As an application, we give a new proof of a result of Rédei-Megyesi about the number of directions determined by a set in a finite affine plane.