<abstract xmlns="http://www.w3.org/1999/xhtml"><p>Let <italic>p ≥</italic> 3 be a prime, <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0012_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>S</mi><mo>⊆</mo><msubsup><mrow><mi>𝔽</mi></mrow><mi>p</mi><mn>2</mn></msubsup></mrow></math><tex-math>S \subseteq \mathbb{F}_p^2</tex-math></alternatives></inline-formula> a nonempty set, and <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0012_eq_003.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>w</mi><mo>:</mo><msubsup><mrow><mi>𝔽</mi></mrow><mi>p</mi><mn>2</mn></msubsup><mo>→</mo><mi>R</mi></mrow></math><tex-math>w:\mathbb{F}_p^2 \to R</tex-math></alternatives></inline-formula> a function with supp <italic>w</italic> = <italic>S</italic>. Applying an uncertainty inequality due to András Bíró and the present author, we show that there are at most <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0012_eq_004.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mrow><mo>|</mo><mi>S</mi><mo>|</mo></mrow></mrow></math><tex-math>{1 \over 2}\left| S \right|</tex-math></alternatives></inline-formula> directions in <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0012_eq_005.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>𝔽</mi></mrow><mi>p</mi><mn>2</mn></msubsup></mrow></math><tex-math>\mathbb{F}_p^2</tex-math></alternatives></inline-formula> such that for every line <italic>l</italic> in any of these directions, one has
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0012_eq_006.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><munder><mo>∑</mo><mrow><mi>z</mi><mo>∈</mo><mi>l</mi></mrow></munder><mrow><mi>w</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mi>p</mi></mfrac><munder><mo>∑</mo><mrow><mi>z</mi><mo>∈</mo><msubsup><mrow><mi>𝔽</mi></mrow><mi>p</mi><mn>2</mn></msubsup></mrow></munder><mrow><mi>w</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>,</mo></mrow></mrow></mrow></math><tex-math>\sum\limits_{z \in l} {w\left( z \right) = {1 \over p}\sum\limits_{z \in \mathbb{F}_p^2} {w\left( z \right),} }</tex-math></alternatives></disp-formula>
except if <italic>S</italic> itself is a line and <italic>w</italic> is constant on <italic>S</italic> (in which case all, but one direction have the property in question). The bound <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0012_eq_007.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mrow><mo>|</mo><mi>S</mi><mo>|</mo></mrow></mrow></math><tex-math>{1 \over 2}\left| S \right|</tex-math></alternatives></inline-formula> is sharp.</p><p>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.</p></abstract>

Let p ≥ 3 be a prime, S⊆𝔽p2S \subseteq \mathbb{F}_p^2 a nonempty set, and w:𝔽p2→Rw:\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
∑z∈lw(z)=1p∑z∈𝔽p2w(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.

Point Distribution and Perfect Directions in <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0012_eq_001.png"/><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>𝔽</mi></mrow><mi>p</mi><mn>2</mn></msubsup></mrow></math><tex-math>\mathbb{F}_p^2</tex-math></alternatives></inline-formula>

Uniform distribution theory

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Point Distribution and Perfect Directions in 

Let p ≥ 3 be a prime, S⊆𝔽p2S \subseteq \mathbb{F}_p^2 a nonempty set, and w:𝔽p2→Rw:\mathbb{F}_p^2 \to R a function with supp w = S. Applying an uncertainty...

Point Distribution and Perfect Directions in $𝔽_{p}^{2}$ \mathbb{F}_p^2

Publié en ligne: 25 déc. 2020

Pages: 93 - 98

Reçu: 08 mars 2019

Accepté: 04 nov. 2020

DOI: https://doi.org/10.2478/udt-2020-0012

Mots clés
Uniform distribution, affine plane

© 2020 Vsevolod F. Lev, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

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

Publié en ligne: 25 déc. 2020

Pages: 93 - 98

Reçu: 08 mars 2019

Accepté: 04 nov. 2020

DOI: https://doi.org/10.2478/udt-2020-0012

Mots clésUniform distribution, affine plane

© 2020 Vsevolod F. Lev, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Point Distribution and Perfect Directions in $𝔽_{p}^{2}$ \mathbb{F}_p^2

Mots clés
Uniform distribution, affine plane