Finding product sets in some classes of amenable groups
In [15], using methods from ergodic theory, a longstanding conjecture of Erdős (see [5, Page 305]) about sumsets in large subsets of the natural numbers was resolved. In this paper, we extend this result to several important classes of amenable groups, including all finitely generated virtually nilp...
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Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509424001555/type/journal_article |
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| author | Dimitrios Charamaras Andreas Mountakis |
| author_facet | Dimitrios Charamaras Andreas Mountakis |
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| description | In [15], using methods from ergodic theory, a longstanding conjecture of Erdős (see [5, Page 305]) about sumsets in large subsets of the natural numbers was resolved. In this paper, we extend this result to several important classes of amenable groups, including all finitely generated virtually nilpotent groups and all abelian groups
$(G,+)$
with the property that the subgroup
$2G := \{g+g : g\in G\}$
has finite index. We prove that in any group G from the above classes, any
$A\subset G$
with positive upper Banach density contains a shifted product set of the form
$\{tb_ib_j\colon i<j\}$
, for some infinite sequence
$(b_n)_{n\in \mathbb {N}}$
and some
$t\in G$
. In fact, we show this result for all amenable groups that posses a property which we call square absolute continuity. Our results provide answers to several questions and conjectures posed in [13]. |
| format | Article |
| id | doaj-art-cf2a2c2391ee49358ab393584df197bb |
| institution | Kabale University |
| issn | 2050-5094 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj-art-cf2a2c2391ee49358ab393584df197bb2025-01-24T05:20:17ZengCambridge University PressForum of Mathematics, Sigma2050-50942025-01-011310.1017/fms.2024.155Finding product sets in some classes of amenable groupsDimitrios Charamaras0https://orcid.org/0000-0001-9574-2362Andreas Mountakis1https://orcid.org/0009-0006-2207-108XInstitut de Mathématiques, École Polytechnique Fédérale de Lausanne (EPFL), EPFL FSB SMA, Station 8, 1015 Lausanne, Switzerland;Department of Mathematics and Applied Mathematics, University of Crete, Voutes Campus, 70013 Heraklion, Greece; E-mail:In [15], using methods from ergodic theory, a longstanding conjecture of Erdős (see [5, Page 305]) about sumsets in large subsets of the natural numbers was resolved. In this paper, we extend this result to several important classes of amenable groups, including all finitely generated virtually nilpotent groups and all abelian groups $(G,+)$ with the property that the subgroup $2G := \{g+g : g\in G\}$ has finite index. We prove that in any group G from the above classes, any $A\subset G$ with positive upper Banach density contains a shifted product set of the form $\{tb_ib_j\colon i<j\}$ , for some infinite sequence $(b_n)_{n\in \mathbb {N}}$ and some $t\in G$ . In fact, we show this result for all amenable groups that posses a property which we call square absolute continuity. Our results provide answers to several questions and conjectures posed in [13].https://www.cambridge.org/core/product/identifier/S2050509424001555/type/journal_article05D1037A1511B1311B30 |
| spellingShingle | Dimitrios Charamaras Andreas Mountakis Finding product sets in some classes of amenable groups Forum of Mathematics, Sigma 05D10 37A15 11B13 11B30 |
| title | Finding product sets in some classes of amenable groups |
| title_full | Finding product sets in some classes of amenable groups |
| title_fullStr | Finding product sets in some classes of amenable groups |
| title_full_unstemmed | Finding product sets in some classes of amenable groups |
| title_short | Finding product sets in some classes of amenable groups |
| title_sort | finding product sets in some classes of amenable groups |
| topic | 05D10 37A15 11B13 11B30 |
| url | https://www.cambridge.org/core/product/identifier/S2050509424001555/type/journal_article |
| work_keys_str_mv | AT dimitrioscharamaras findingproductsetsinsomeclassesofamenablegroups AT andreasmountakis findingproductsetsinsomeclassesofamenablegroups |