1. bookVolumen 15 (2020): Heft 2 (December 2020)
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Zeitschrift
eISSN
2309-5377
Erstveröffentlichung
30 Dec 2013
Erscheinungsweise
2 Hefte pro Jahr
Sprachen
Englisch
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Point Distribution and Perfect Directions in 𝔽p2\mathbb{F}_p^2

Online veröffentlicht: 25 Dec 2020
Volumen & Heft: Volumen 15 (2020) - Heft 2 (December 2020)
Seitenbereich: 93 - 98
Eingereicht: 08 Mar 2019
Akzeptiert: 04 Nov 2020
Zeitschriftendaten
License
Format
Zeitschrift
eISSN
2309-5377
Erstveröffentlichung
30 Dec 2013
Erscheinungsweise
2 Hefte pro Jahr
Sprachen
Englisch
Abstract

Let p ≥ 3 be a prime, S𝔽p2S \subseteq \mathbb{F}_p^2 a nonempty set, and w:𝔽p2Rw:\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 12|S|{1 \over 2}\left| S \right| directions in 𝔽p2\mathbb{F}_p^2 such that for every line l in any of these directions, one has zlw(z)=1pz𝔽p2w(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 12|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.

MSC 2010

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