Number of Directions Determined by a Set in [Formula: see text] and Growth in [Formula: see text]

We prove that a set A of at most q non-collinear points in the finite plane [Formula: see text] spans more than [Formula: see text] directions: this is based on a lower bound by Fancsali et al. which we prove again together with a different upper bound than the one given therein. Then, following the...

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Autor principal: Dona, Daniele
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Springer US 2021
Materias:
Article
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8558295/
https://www.ncbi.nlm.nih.gov/pubmed/34789956
http://dx.doi.org/10.1007/s00454-021-00284-6
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author Dona, Daniele
author_facet Dona, Daniele
author_sort Dona, Daniele
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description We prove that a set A of at most q non-collinear points in the finite plane [Formula: see text] spans more than [Formula: see text] directions: this is based on a lower bound by Fancsali et al. which we prove again together with a different upper bound than the one given therein. Then, following the procedure used by Rudnev and Shkredov, we prove a new structural theorem about slowly growing sets in [Formula: see text] for any finite field [Formula: see text] , generalizing the analogous results by Helfgott, Murphy, and Rudnev and Shkredov over prime fields.
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spelling pubmed-85582952021-11-15 Number of Directions Determined by a Set in [Formula: see text] and Growth in [Formula: see text] Dona, Daniele Discrete Comput Geom Article We prove that a set A of at most q non-collinear points in the finite plane [Formula: see text] spans more than [Formula: see text] directions: this is based on a lower bound by Fancsali et al. which we prove again together with a different upper bound than the one given therein. Then, following the procedure used by Rudnev and Shkredov, we prove a new structural theorem about slowly growing sets in [Formula: see text] for any finite field [Formula: see text] , generalizing the analogous results by Helfgott, Murphy, and Rudnev and Shkredov over prime fields. Springer US 2021-03-08 2021 /pmc/articles/PMC8558295/ /pubmed/34789956 http://dx.doi.org/10.1007/s00454-021-00284-6 Text en © The Author(s) 2021 https://creativecommons.org/licenses/by/4.0/Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/ (https://creativecommons.org/licenses/by/4.0/) .
spellingShingle Article
Dona, Daniele
Number of Directions Determined by a Set in [Formula: see text] and Growth in [Formula: see text]
title Number of Directions Determined by a Set in [Formula: see text] and Growth in [Formula: see text]
title_full Number of Directions Determined by a Set in [Formula: see text] and Growth in [Formula: see text]
title_fullStr Number of Directions Determined by a Set in [Formula: see text] and Growth in [Formula: see text]
title_full_unstemmed Number of Directions Determined by a Set in [Formula: see text] and Growth in [Formula: see text]
title_short Number of Directions Determined by a Set in [Formula: see text] and Growth in [Formula: see text]
title_sort number of directions determined by a set in [formula: see text] and growth in [formula: see text]
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8558295/
https://www.ncbi.nlm.nih.gov/pubmed/34789956
http://dx.doi.org/10.1007/s00454-021-00284-6
work_keys_str_mv AT donadaniele numberofdirectionsdeterminedbyasetinformulaseetextandgrowthinformulaseetext

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