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Large subsets of Z m n without arithmetic progressions
For integers and , we study the problem of finding good lower bounds for the size of progression-free sets in . Let denote the maximal size of a subset of without arithmetic progressions of length and let denote the least prime factor of . We construct explicit progression-free sets and obtain the...
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| Published in: | Designs, codes, and cryptography codes, and cryptography, 2023, Vol.91 (4), p.1443 |
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| Main Authors: | , , |
| Format: | Article |
| Language: | English |
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| container_issue | 4 |
| container_start_page | 1443 |
| container_title | Designs, codes, and cryptography |
| container_volume | 91 |
| creator | Elsholtz, Christian Klahn, Benjamin Lipnik, Gabriel F |
| description | For integers
and
, we study the problem of finding good lower bounds for the size of progression-free sets in
. Let
denote the maximal size of a subset of
without arithmetic progressions of length
and let
denote the least prime factor of
. We construct explicit progression-free sets and obtain the following improved lower bounds for
:If
is odd and
, then
If
is even,
and
, then
Moreover, we give some further improved lower bounds on
for primes
and progression lengths
. |
| doi_str_mv | 10.1007/s10623-022-01145-w |
| format | article |
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and
, we study the problem of finding good lower bounds for the size of progression-free sets in
. Let
denote the maximal size of a subset of
without arithmetic progressions of length
and let
denote the least prime factor of
. We construct explicit progression-free sets and obtain the following improved lower bounds for
:If
is odd and
, then
If
is even,
and
, then
Moreover, we give some further improved lower bounds on
for primes
and progression lengths
.</description><identifier>ISSN: 1573-7586</identifier><identifier>EISSN: 1573-7586</identifier><identifier>DOI: 10.1007/s10623-022-01145-w</identifier><identifier>PMID: 37035093</identifier><language>eng</language><publisher>United States</publisher><ispartof>Designs, codes, and cryptography, 2023, Vol.91 (4), p.1443</ispartof><rights>The Author(s) 2022.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><orcidid>0000-0002-2960-4030 ; 0000-0002-4362-429X</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,780,784,4024,27923,27924,27925</link.rule.ids><backlink>$$Uhttps://www.ncbi.nlm.nih.gov/pubmed/37035093$$D View this record in MEDLINE/PubMed$$Hfree_for_read</backlink></links><search><creatorcontrib>Elsholtz, Christian</creatorcontrib><creatorcontrib>Klahn, Benjamin</creatorcontrib><creatorcontrib>Lipnik, Gabriel F</creatorcontrib><title>Large subsets of Z m n without arithmetic progressions</title><title>Designs, codes, and cryptography</title><addtitle>Des Codes Cryptogr</addtitle><description>For integers
and
, we study the problem of finding good lower bounds for the size of progression-free sets in
. Let
denote the maximal size of a subset of
without arithmetic progressions of length
and let
denote the least prime factor of
. We construct explicit progression-free sets and obtain the following improved lower bounds for
:If
is odd and
, then
If
is even,
and
, then
Moreover, we give some further improved lower bounds on
for primes
and progression lengths
.</description><issn>1573-7586</issn><issn>1573-7586</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2023</creationdate><recordtype>article</recordtype><recordid>eNpNj81Kw0AUhQdRbK2-gAuZpZvRO3M7f0sp_kHATVduwk17UyNNUjMJxbfxWXwyA1Zwdb7FxzkcIS413GgAf5s0OIMKjFGg9dyq_ZGYautReRvc8T-eiLOU3gFAI5hTMUEPaCHiVISMug3LNBSJ-yTb8vtLvspaNnJf9W_t0EvqRqi5r1Zy17WbjlOq2iadi5OStokvDjkTy4f75eJJZS-Pz4u7TO2sQ0VYrDgAE2myJRgH6KNnGyKZMnrNEcmZNcRYluSpgMBu7j17xxTBBJyJ69_acftj4NTndZVWvN1Sw-2QcuNj1N4FjKN6dVCHouZ1vuuqmrrP_O8s_gD72FbR</recordid><startdate>2023</startdate><enddate>2023</enddate><creator>Elsholtz, Christian</creator><creator>Klahn, Benjamin</creator><creator>Lipnik, Gabriel F</creator><scope>NPM</scope><scope>7X8</scope><orcidid>https://orcid.org/0000-0002-2960-4030</orcidid><orcidid>https://orcid.org/0000-0002-4362-429X</orcidid></search><sort><creationdate>2023</creationdate><title>Large subsets of Z m n without arithmetic progressions</title><author>Elsholtz, Christian ; Klahn, Benjamin ; Lipnik, Gabriel F</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-p563-a3bce80eaa1a5f02603797e589a2f971e93a62d099ffa7ab08e6477e76ea90283</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2023</creationdate><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Elsholtz, Christian</creatorcontrib><creatorcontrib>Klahn, Benjamin</creatorcontrib><creatorcontrib>Lipnik, Gabriel F</creatorcontrib><collection>PubMed</collection><collection>MEDLINE - Academic</collection><jtitle>Designs, codes, and cryptography</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Elsholtz, Christian</au><au>Klahn, Benjamin</au><au>Lipnik, Gabriel F</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Large subsets of Z m n without arithmetic progressions</atitle><jtitle>Designs, codes, and cryptography</jtitle><addtitle>Des Codes Cryptogr</addtitle><date>2023</date><risdate>2023</risdate><volume>91</volume><issue>4</issue><spage>1443</spage><pages>1443-</pages><issn>1573-7586</issn><eissn>1573-7586</eissn><abstract>For integers
and
, we study the problem of finding good lower bounds for the size of progression-free sets in
. Let
denote the maximal size of a subset of
without arithmetic progressions of length
and let
denote the least prime factor of
. We construct explicit progression-free sets and obtain the following improved lower bounds for
:If
is odd and
, then
If
is even,
and
, then
Moreover, we give some further improved lower bounds on
for primes
and progression lengths
.</abstract><cop>United States</cop><pmid>37035093</pmid><doi>10.1007/s10623-022-01145-w</doi><orcidid>https://orcid.org/0000-0002-2960-4030</orcidid><orcidid>https://orcid.org/0000-0002-4362-429X</orcidid></addata></record> |
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| ispartof | Designs, codes, and cryptography, 2023, Vol.91 (4), p.1443 |
| issn | 1573-7586 1573-7586 |
| language | eng |
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| source | Springer Nature |
| title | Large subsets of Z m n without arithmetic progressions |
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