Large subsets of [Formula: see text] without arithmetic progressions

For integers m and n, we study the problem of finding good lower bounds for the size of progression-free sets in [Formula: see text] . Let [Formula: see text] denote the maximal size of a subset of [Formula: see text] without arithmetic progressions of length k and let [Formula: see text] denote the...

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Autores principales: Elsholtz, Christian, Klahn, Benjamin, Lipnik, Gabriel F.
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Springer US 2022
Materias:
Article
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10073169/
https://www.ncbi.nlm.nih.gov/pubmed/37035093
http://dx.doi.org/10.1007/s10623-022-01145-w
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author Elsholtz, Christian
Klahn, Benjamin
Lipnik, Gabriel F.
author_facet Elsholtz, Christian
Klahn, Benjamin
Lipnik, Gabriel F.
author_sort Elsholtz, Christian
collection PubMed
description For integers m and n, we study the problem of finding good lower bounds for the size of progression-free sets in [Formula: see text] . Let [Formula: see text] denote the maximal size of a subset of [Formula: see text] without arithmetic progressions of length k and let [Formula: see text] denote the least prime factor of m. We construct explicit progression-free sets and obtain the following improved lower bounds for [Formula: see text] : If [Formula: see text] is odd and [Formula: see text] , then [Formula: see text] If [Formula: see text] is even, [Formula: see text] and [Formula: see text] , then [Formula: see text] Moreover, we give some further improved lower bounds on [Formula: see text] for primes [Formula: see text] and progression lengths [Formula: see text] .
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spelling pubmed-100731692023-04-06 Large subsets of [Formula: see text] without arithmetic progressions Elsholtz, Christian Klahn, Benjamin Lipnik, Gabriel F. Des Codes Cryptogr Article For integers m and n, we study the problem of finding good lower bounds for the size of progression-free sets in [Formula: see text] . Let [Formula: see text] denote the maximal size of a subset of [Formula: see text] without arithmetic progressions of length k and let [Formula: see text] denote the least prime factor of m. We construct explicit progression-free sets and obtain the following improved lower bounds for [Formula: see text] : If [Formula: see text] is odd and [Formula: see text] , then [Formula: see text] If [Formula: see text] is even, [Formula: see text] and [Formula: see text] , then [Formula: see text] Moreover, we give some further improved lower bounds on [Formula: see text] for primes [Formula: see text] and progression lengths [Formula: see text] . Springer US 2022-12-15 2023 /pmc/articles/PMC10073169/ /pubmed/37035093 http://dx.doi.org/10.1007/s10623-022-01145-w Text en © The Author(s) 2022 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
Elsholtz, Christian
Klahn, Benjamin
Lipnik, Gabriel F.
Large subsets of [Formula: see text] without arithmetic progressions
title Large subsets of [Formula: see text] without arithmetic progressions
title_full Large subsets of [Formula: see text] without arithmetic progressions
title_fullStr Large subsets of [Formula: see text] without arithmetic progressions
title_full_unstemmed Large subsets of [Formula: see text] without arithmetic progressions
title_short Large subsets of [Formula: see text] without arithmetic progressions
title_sort large subsets of [formula: see text] without arithmetic progressions
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10073169/
https://www.ncbi.nlm.nih.gov/pubmed/37035093
http://dx.doi.org/10.1007/s10623-022-01145-w
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