On the arithmetic Kakeya conjecture of Katz and Tao

The arithmetic Kakeya conjecture, formulated by Katz and Tao (Math Res Lett 6(5–6):625–630, 1999), is a statement about addition of finite sets. It is known to imply a form of the Kakeya conjecture, namely that the upper Minkowski dimension of a Besicovitch set in [Formula: see text] is n. In this n...

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Detalles Bibliográficos
Autores principales: Green, Ben, Ruzsa, Imre Z.
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Springer International Publishing 2018
Materias:
Article
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6528801/
https://www.ncbi.nlm.nih.gov/pubmed/31178607
http://dx.doi.org/10.1007/s10998-018-0270-z
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author Green, Ben
Ruzsa, Imre Z.
author_facet Green, Ben
Ruzsa, Imre Z.
author_sort Green, Ben
collection PubMed
description The arithmetic Kakeya conjecture, formulated by Katz and Tao (Math Res Lett 6(5–6):625–630, 1999), is a statement about addition of finite sets. It is known to imply a form of the Kakeya conjecture, namely that the upper Minkowski dimension of a Besicovitch set in [Formula: see text] is n. In this note we discuss this conjecture, giving a number of equivalent forms of it. We show that a natural finite field variant of it does hold. We also give some lower bounds.
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spelling pubmed-65288012019-06-07 On the arithmetic Kakeya conjecture of Katz and Tao Green, Ben Ruzsa, Imre Z. Period Math Hung Article The arithmetic Kakeya conjecture, formulated by Katz and Tao (Math Res Lett 6(5–6):625–630, 1999), is a statement about addition of finite sets. It is known to imply a form of the Kakeya conjecture, namely that the upper Minkowski dimension of a Besicovitch set in [Formula: see text] is n. In this note we discuss this conjecture, giving a number of equivalent forms of it. We show that a natural finite field variant of it does hold. We also give some lower bounds. Springer International Publishing 2018-11-02 2019 /pmc/articles/PMC6528801/ /pubmed/31178607 http://dx.doi.org/10.1007/s10998-018-0270-z Text en © The Author(s) 2018 Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
spellingShingle Article
Green, Ben
Ruzsa, Imre Z.
On the arithmetic Kakeya conjecture of Katz and Tao
title On the arithmetic Kakeya conjecture of Katz and Tao
title_full On the arithmetic Kakeya conjecture of Katz and Tao
title_fullStr On the arithmetic Kakeya conjecture of Katz and Tao
title_full_unstemmed On the arithmetic Kakeya conjecture of Katz and Tao
title_short On the arithmetic Kakeya conjecture of Katz and Tao
title_sort on the arithmetic kakeya conjecture of katz and tao
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6528801/
https://www.ncbi.nlm.nih.gov/pubmed/31178607
http://dx.doi.org/10.1007/s10998-018-0270-z
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