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On some maximal and minimal sets
A set A of positive integers is called 3-free if it contains no 3-term arithmetic progression. Furthermore, such A is called maximal if it is not properly contained in any other 3-free set. In 2006, by confirming a question posed by Erdős et al., Savchev and Chen proved that there exists a maximal 3...
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| Published in: | Periodica mathematica Hungarica 2024-06, Vol.88 (2), p.354-358 |
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| container_end_page | 358 |
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| container_title | Periodica mathematica Hungarica |
| container_volume | 88 |
| creator | Fang, Jin-Hui Cao, Xue-Qin |
| description | A set
A
of positive integers is called 3-free if it contains no 3-term arithmetic progression. Furthermore, such
A
is called
maximal
if it is not properly contained in any other 3-free set. In 2006, by confirming a question posed by Erdős et al., Savchev and Chen proved that there exists a maximal 3-free set
{
a
1
<
a
2
<
⋯
<
a
n
<
⋯
}
of positive integers with the property that
lim
n
→
∞
(
a
n
+
1
-
a
n
)
=
∞
. In this paper, we generalize their result. On the other hand, a set
A
of nonnegative integers is called an asymptotic basis of order
h
if every sufficiently large integer can be represented as a sum of
h
elements of
A
. Such
A
is defined as
minimal
if no proper subset of
A
has this property. We also extend a result of Jańczak and Schoen about the minimal asymptotic basis. |
| doi_str_mv | 10.1007/s10998-023-00559-w |
| format | article |
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A
of positive integers is called 3-free if it contains no 3-term arithmetic progression. Furthermore, such
A
is called
maximal
if it is not properly contained in any other 3-free set. In 2006, by confirming a question posed by Erdős et al., Savchev and Chen proved that there exists a maximal 3-free set
{
a
1
<
a
2
<
⋯
<
a
n
<
⋯
}
of positive integers with the property that
lim
n
→
∞
(
a
n
+
1
-
a
n
)
=
∞
. In this paper, we generalize their result. On the other hand, a set
A
of nonnegative integers is called an asymptotic basis of order
h
if every sufficiently large integer can be represented as a sum of
h
elements of
A
. Such
A
is defined as
minimal
if no proper subset of
A
has this property. We also extend a result of Jańczak and Schoen about the minimal asymptotic basis.</description><identifier>ISSN: 0031-5303</identifier><identifier>EISSN: 1588-2829</identifier><identifier>DOI: 10.1007/s10998-023-00559-w</identifier><language>eng</language><publisher>Cham: Springer International Publishing</publisher><subject>Asymptotic properties ; Integers ; Mathematics ; Mathematics and Statistics</subject><ispartof>Periodica mathematica Hungarica, 2024-06, Vol.88 (2), p.354-358</ispartof><rights>Akadémiai Kiadó, Budapest, Hungary 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c270t-81f25ae4c10087c3b9f76b0926faaa83f890b26b622ca79278bb87a90943e6b63</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,780,784,27924,27925</link.rule.ids></links><search><creatorcontrib>Fang, Jin-Hui</creatorcontrib><creatorcontrib>Cao, Xue-Qin</creatorcontrib><title>On some maximal and minimal sets</title><title>Periodica mathematica Hungarica</title><addtitle>Period Math Hung</addtitle><description>A set
A
of positive integers is called 3-free if it contains no 3-term arithmetic progression. Furthermore, such
A
is called
maximal
if it is not properly contained in any other 3-free set. In 2006, by confirming a question posed by Erdős et al., Savchev and Chen proved that there exists a maximal 3-free set
{
a
1
<
a
2
<
⋯
<
a
n
<
⋯
}
of positive integers with the property that
lim
n
→
∞
(
a
n
+
1
-
a
n
)
=
∞
. In this paper, we generalize their result. On the other hand, a set
A
of nonnegative integers is called an asymptotic basis of order
h
if every sufficiently large integer can be represented as a sum of
h
elements of
A
. Such
A
is defined as
minimal
if no proper subset of
A
has this property. We also extend a result of Jańczak and Schoen about the minimal asymptotic basis.</description><subject>Asymptotic properties</subject><subject>Integers</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><issn>0031-5303</issn><issn>1588-2829</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><recordid>eNp9kEFLAzEQhYMouK7-AU8LnqOTme4mOUpRKxR60XNI1kRaurs12VL998au4E0YmGF435vhMXYt4FYAyLskQGvFAYkD1LXmhxNWiFopjgr1KSsASPCagM7ZRUobgIwRFKxa9VUaOl919nPd2W1l-7eqW_fHOfkxXbKzYLfJX_32kr0-PrzMF3y5enqe3y95ixJGrkTA2vpZm32VbMnpIBsHGptgrVUUlAaHjWsQWys1SuWcklaDnpHPayrZzeS7i8PH3qfRbIZ97PNJQ9CA1EC5SoaTqo1DStEHs4v51fhlBJifJMyUhMlJmGMS5pAhmqCUxf27j3_W_1DfBjxfBw</recordid><startdate>20240601</startdate><enddate>20240601</enddate><creator>Fang, Jin-Hui</creator><creator>Cao, Xue-Qin</creator><general>Springer International Publishing</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20240601</creationdate><title>On some maximal and minimal sets</title><author>Fang, Jin-Hui ; Cao, Xue-Qin</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c270t-81f25ae4c10087c3b9f76b0926faaa83f890b26b622ca79278bb87a90943e6b63</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Asymptotic properties</topic><topic>Integers</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Fang, Jin-Hui</creatorcontrib><creatorcontrib>Cao, Xue-Qin</creatorcontrib><collection>CrossRef</collection><jtitle>Periodica mathematica Hungarica</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Fang, Jin-Hui</au><au>Cao, Xue-Qin</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On some maximal and minimal sets</atitle><jtitle>Periodica mathematica Hungarica</jtitle><stitle>Period Math Hung</stitle><date>2024-06-01</date><risdate>2024</risdate><volume>88</volume><issue>2</issue><spage>354</spage><epage>358</epage><pages>354-358</pages><issn>0031-5303</issn><eissn>1588-2829</eissn><abstract>A set
A
of positive integers is called 3-free if it contains no 3-term arithmetic progression. Furthermore, such
A
is called
maximal
if it is not properly contained in any other 3-free set. In 2006, by confirming a question posed by Erdős et al., Savchev and Chen proved that there exists a maximal 3-free set
{
a
1
<
a
2
<
⋯
<
a
n
<
⋯
}
of positive integers with the property that
lim
n
→
∞
(
a
n
+
1
-
a
n
)
=
∞
. In this paper, we generalize their result. On the other hand, a set
A
of nonnegative integers is called an asymptotic basis of order
h
if every sufficiently large integer can be represented as a sum of
h
elements of
A
. Such
A
is defined as
minimal
if no proper subset of
A
has this property. We also extend a result of Jańczak and Schoen about the minimal asymptotic basis.</abstract><cop>Cham</cop><pub>Springer International Publishing</pub><doi>10.1007/s10998-023-00559-w</doi><tpages>5</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0031-5303 |
| ispartof | Periodica mathematica Hungarica, 2024-06, Vol.88 (2), p.354-358 |
| issn | 0031-5303 1588-2829 |
| language | eng |
| recordid | cdi_proquest_journals_3060790390 |
| source | Springer Nature |
| subjects | Asymptotic properties Integers Mathematics Mathematics and Statistics |
| title | On some maximal and minimal sets |
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