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

Lev, Vsevolod F.

doi:10.2478/udt-2020-0012

Abstract

Let p ≥ 3 be a prime, $S \subseteq 𝔽_{p}^{2}$ S \subseteq \mathbb{F}_p^2 a nonempty set, and $w : 𝔽_{p}^{2} \to R$ 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 $\frac{1}{2} | S |$ {1 \over 2}\left| S \right| directions in $𝔽_{p}^{2}$ \mathbb{F}_p^2 such that for every line l in any of these directions, one has $\sum_{z \in l} w (z) = \frac{1}{p} \sum_{z \in 𝔽_{p}^{2}} w (z),$ \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 $\frac{1}{2} | S |$ {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.

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

Abstract

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