Infinite sumsets in sets with positive density
Motivated by questions asked by Erdős, we prove that any set A ⊂ N A\subset \mathbb {N} with positive upper density contains, for any k ∈ N k\in \mathbb {N} , a sumset B 1 + ⋯ + B k B_1+\cdots +B_k , where B 1 B_1 , …, B k ⊂ N B_k\subset \mathbb {N} are infinite. Our proof uses ergodic theory and re...
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| Veröffentlicht in: | Journal of the American Mathematical Society 2024-07, Vol.37 (3), p.637-682 |
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| Sprache: | eng |
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| container_title | Journal of the American Mathematical Society |
| container_volume | 37 |
| creator | Kra, Bryna Moreira, Joel Richter, Florian Robertson, Donald |
| description | Motivated by questions asked by Erdős, we prove that any set
A
⊂
N
A\subset \mathbb {N}
with positive upper density contains, for any
k
∈
N
k\in \mathbb {N}
, a sumset
B
1
+
⋯
+
B
k
B_1+\cdots +B_k
, where
B
1
B_1
, …,
B
k
⊂
N
B_k\subset \mathbb {N}
are infinite. Our proof uses ergodic theory and relies on structural results for measure preserving systems. Our techniques are new, even for the previously known case of
k
=
2
k=2
. |
| doi_str_mv | 10.1090/jams/1030 |
| format | Article |
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A
⊂
N
A\subset \mathbb {N}
with positive upper density contains, for any
k
∈
N
k\in \mathbb {N}
, a sumset
B
1
+
⋯
+
B
k
B_1+\cdots +B_k
, where
B
1
B_1
, …,
B
k
⊂
N
B_k\subset \mathbb {N}
are infinite. Our proof uses ergodic theory and relies on structural results for measure preserving systems. Our techniques are new, even for the previously known case of
k
=
2
k=2
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A
⊂
N
A\subset \mathbb {N}
with positive upper density contains, for any
k
∈
N
k\in \mathbb {N}
, a sumset
B
1
+
⋯
+
B
k
B_1+\cdots +B_k
, where
B
1
B_1
, …,
B
k
⊂
N
B_k\subset \mathbb {N}
are infinite. Our proof uses ergodic theory and relies on structural results for measure preserving systems. Our techniques are new, even for the previously known case of
k
=
2
k=2
.</description><issn>0894-0347</issn><issn>1088-6834</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><recordid>eNotzz1PwzAUhWELUYnQMvAPsjK4vf6Icz2iio9KlVhgthz3WrgiaRUbUP89BDq9ZzrSw9itgKUAC6u97_NKgIILVglA5AaVvmQVoNUclG6v2HXOewAQpjEVW26GmIZUqM6ffaaS6zTUf_1O5b0-HnIq6YvqHQ2_67Rgs-g_Mt2cO2dvjw-v62e-fXnarO-3PIjGFo4guoitRB8ktr4h1EZgACsjWrI6BknYNdq30mspUQmz8x3EAB0ajaTm7O7_N4yHnEeK7jim3o8nJ8BNUDdB3QRVP-6ZRf4</recordid><startdate>202407</startdate><enddate>202407</enddate><creator>Kra, Bryna</creator><creator>Moreira, Joel</creator><creator>Richter, Florian</creator><creator>Robertson, Donald</creator><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>202407</creationdate><title>Infinite sumsets in sets with positive density</title><author>Kra, Bryna ; Moreira, Joel ; Richter, Florian ; Robertson, Donald</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c159t-801bf8728ac287a5e84618c092f89e94fc2e8b54a72a4228316dab0fc0b8648e3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Kra, Bryna</creatorcontrib><creatorcontrib>Moreira, Joel</creatorcontrib><creatorcontrib>Richter, Florian</creatorcontrib><creatorcontrib>Robertson, Donald</creatorcontrib><collection>CrossRef</collection><jtitle>Journal of the American Mathematical Society</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Kra, Bryna</au><au>Moreira, Joel</au><au>Richter, Florian</au><au>Robertson, Donald</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Infinite sumsets in sets with positive density</atitle><jtitle>Journal of the American Mathematical Society</jtitle><date>2024-07</date><risdate>2024</risdate><volume>37</volume><issue>3</issue><spage>637</spage><epage>682</epage><pages>637-682</pages><issn>0894-0347</issn><eissn>1088-6834</eissn><abstract>Motivated by questions asked by Erdős, we prove that any set
A
⊂
N
A\subset \mathbb {N}
with positive upper density contains, for any
k
∈
N
k\in \mathbb {N}
, a sumset
B
1
+
⋯
+
B
k
B_1+\cdots +B_k
, where
B
1
B_1
, …,
B
k
⊂
N
B_k\subset \mathbb {N}
are infinite. Our proof uses ergodic theory and relies on structural results for measure preserving systems. Our techniques are new, even for the previously known case of
k
=
2
k=2
.</abstract><doi>10.1090/jams/1030</doi><tpages>46</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0894-0347 |
| ispartof | Journal of the American Mathematical Society, 2024-07, Vol.37 (3), p.637-682 |
| issn | 0894-0347 1088-6834 |
| language | eng |
| recordid | cdi_crossref_primary_10_1090_jams_1030 |
| source | American Mathematical Society Publications |
| title | Infinite sumsets in sets with positive density |
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