Teoria ergódica de Ramsey e a conjectura B + B + t

Detalhes bibliográficos
Ano de defesa: 2025
Autor(a) principal: Santos Júnior, Jônatas Marinho
Orientador(a): Lima, Yuri Gomes
Banca de defesa: Não Informado pela instituição
Tipo de documento: Dissertação
Tipo de acesso: Acesso aberto
Idioma: por
Instituição de defesa: Não Informado pela instituição
Programa de Pós-Graduação: Não Informado pela instituição
Departamento: Não Informado pela instituição
País: Não Informado pela instituição
Área do conhecimento CNPq:
CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA::GEOMETRIA E TOPOLOGIA::SISTEMAS DINAMICOS
Link de acesso: http://repositorio.ufc.br/handle/riufc/81643
Resumo: In this work, we explore the relation between two distinct areas of Mathematics, namely Ergodic Theory and Ramsey Theory. The interaction between these two areas is nowadays called Ergodic Ramsey Theory. We will present the proof of two classical results: Roth theorem, where we will use tools from ergodic theory and spectral theory, and Hindman theorem, where we will use ultrafilters from set theory. Finally, we study the proof of Erdős B + B + t Conjecture, that was recently obtained using Ergodic Ramsey Theory. An element of great importance in this dissertation, that will be used both in the proof of Roth theorem and in the proof of B + B + t Conjecture, is the Kronecker factor, that captures an structure of rotation group inside a system that preserves measure.
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spelling Santos Júnior, Jônatas MarinhoLima, Yuri Gomes2025-07-21T13:48:56Z2025-07-21T13:48:56Z2025SANTOS JÚNIOR, Jônatas Marinho. Teoria ergódica de Ramsey e a conjectura B + B + t. 2025. 84 f. Dissertação (Mestrado Acadêmico em Matemática) – Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2025.http://repositorio.ufc.br/handle/riufc/81643In this work, we explore the relation between two distinct areas of Mathematics, namely Ergodic Theory and Ramsey Theory. The interaction between these two areas is nowadays called Ergodic Ramsey Theory. We will present the proof of two classical results: Roth theorem, where we will use tools from ergodic theory and spectral theory, and Hindman theorem, where we will use ultrafilters from set theory. Finally, we study the proof of Erdős B + B + t Conjecture, that was recently obtained using Ergodic Ramsey Theory. An element of great importance in this dissertation, that will be used both in the proof of Roth theorem and in the proof of B + B + t Conjecture, is the Kronecker factor, that captures an structure of rotation group inside a system that preserves measure.Nesta dissertação, exploramos a relação entre duas áreas distintas da Matemática, a Teoria Ergódica e a Teoria de Ramsey. A interação destas áreas é atualmente conhecida como Teoria Ergódica de Ramsey. Apresentaremos a prova de dois resultados clássicos: o Teorema de Roth, onde utilizaremos ferramentas de teoria ergódica e teoria espectral, e o Teorema de Hindman, onde utilizaremos ultrafiltros da teoria de conjuntos. Por fim, estudamos a prova da Conjectura B + B + t de Erdős, que recentemente obtida utilizando Teoria Ergódica de Ramsey. Um elemento de relevante importância nesta dissertação, e que será utilizado nas provas do Teorema de Roth e da Conjectura B + B + t , é o Fator de Kronecker, que encapsula uma estrutura de grupo de rotação dentro de um sistema que preserva medida.Teoria ergódica de Ramsey e a conjectura B + B + tRamsey's ergodic theory and the B+B+t conjectureinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/masterThesisTeoria ergódica de RamseyFator de KroneckerTeorema de RothConjectura B + B + tErgodic Ramsey theoryKronecker factorRoth theoremHindman theoremCNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA::GEOMETRIA E TOPOLOGIA::SISTEMAS DINAMICOSinfo:eu-repo/semantics/openAccessporreponame:Repositório Institucional da Universidade Federal do Ceará (UFC)instname:Universidade Federal do Ceará (UFC)instacron:UFChttps://orcid.org/0009-0004-1349-6778http://lattes.cnpq.br/7446946385040026https://orcid.org/0000-0002-2179-9806https://lattes.cnpq.br/49128474416327802025-07-18ORIGINAL2025_dis_jmsantosjunior.pdf2025_dis_jmsantosjunior.pdfdissertaçao jonatas marinhoapplication/pdf1025001http://repositorio.ufc.br/bitstream/riufc/81643/3/2025_dis_jmsantosjunior.pdff49ea46fdb1ecbd2f76706daa93b5e8fMD53LICENSElicense.txtlicense.txttext/plain; charset=utf-81748http://repositorio.ufc.br/bitstream/riufc/81643/4/license.txt8a4605be74aa9ea9d79846c1fba20a33MD54riufc/816432025-07-21 10:49:06.25oai:repositorio.ufc.br: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Repositório InstitucionalPUBhttp://www.repositorio.ufc.br/ri-oai/requestbu@ufc.br || repositorio@ufc.bropendoar:2025-07-21T13:49:06Repositório Institucional da Universidade Federal do Ceará (UFC) - Universidade Federal do Ceará (UFC)false
dc.title.pt_BR.fl_str_mv Teoria ergódica de Ramsey e a conjectura B + B + t
dc.title.en.pt_BR.fl_str_mv Ramsey's ergodic theory and the B+B+t conjecture
title Teoria ergódica de Ramsey e a conjectura B + B + t
spellingShingle Teoria ergódica de Ramsey e a conjectura B + B + t
Santos Júnior, Jônatas Marinho
CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA::GEOMETRIA E TOPOLOGIA::SISTEMAS DINAMICOS
Teoria ergódica de Ramsey
Fator de Kronecker
Teorema de Roth
Conjectura B + B + t
Ergodic Ramsey theory
Kronecker factor
Roth theorem
Hindman theorem
title_short Teoria ergódica de Ramsey e a conjectura B + B + t
title_full Teoria ergódica de Ramsey e a conjectura B + B + t
title_fullStr Teoria ergódica de Ramsey e a conjectura B + B + t
title_full_unstemmed Teoria ergódica de Ramsey e a conjectura B + B + t
title_sort Teoria ergódica de Ramsey e a conjectura B + B + t
author Santos Júnior, Jônatas Marinho
author_facet Santos Júnior, Jônatas Marinho
author_role author
dc.contributor.author.fl_str_mv Santos Júnior, Jônatas Marinho
dc.contributor.advisor1.fl_str_mv Lima, Yuri Gomes
contributor_str_mv Lima, Yuri Gomes
dc.subject.cnpq.fl_str_mv CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA::GEOMETRIA E TOPOLOGIA::SISTEMAS DINAMICOS
topic CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA::GEOMETRIA E TOPOLOGIA::SISTEMAS DINAMICOS
Teoria ergódica de Ramsey
Fator de Kronecker
Teorema de Roth
Conjectura B + B + t
Ergodic Ramsey theory
Kronecker factor
Roth theorem
Hindman theorem
dc.subject.ptbr.pt_BR.fl_str_mv Teoria ergódica de Ramsey
Fator de Kronecker
Teorema de Roth
Conjectura B + B + t
dc.subject.en.pt_BR.fl_str_mv Ergodic Ramsey theory
Kronecker factor
Roth theorem
Hindman theorem
description In this work, we explore the relation between two distinct areas of Mathematics, namely Ergodic Theory and Ramsey Theory. The interaction between these two areas is nowadays called Ergodic Ramsey Theory. We will present the proof of two classical results: Roth theorem, where we will use tools from ergodic theory and spectral theory, and Hindman theorem, where we will use ultrafilters from set theory. Finally, we study the proof of Erdős B + B + t Conjecture, that was recently obtained using Ergodic Ramsey Theory. An element of great importance in this dissertation, that will be used both in the proof of Roth theorem and in the proof of B + B + t Conjecture, is the Kronecker factor, that captures an structure of rotation group inside a system that preserves measure.
publishDate 2025
dc.date.accessioned.fl_str_mv 2025-07-21T13:48:56Z
dc.date.available.fl_str_mv 2025-07-21T13:48:56Z
dc.date.issued.fl_str_mv 2025
dc.type.status.fl_str_mv info:eu-repo/semantics/publishedVersion
dc.type.driver.fl_str_mv info:eu-repo/semantics/masterThesis
format masterThesis
status_str publishedVersion
dc.identifier.citation.fl_str_mv SANTOS JÚNIOR, Jônatas Marinho. Teoria ergódica de Ramsey e a conjectura B + B + t. 2025. 84 f. Dissertação (Mestrado Acadêmico em Matemática) – Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2025.
dc.identifier.uri.fl_str_mv http://repositorio.ufc.br/handle/riufc/81643
identifier_str_mv SANTOS JÚNIOR, Jônatas Marinho. Teoria ergódica de Ramsey e a conjectura B + B + t. 2025. 84 f. Dissertação (Mestrado Acadêmico em Matemática) – Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2025.
url http://repositorio.ufc.br/handle/riufc/81643
dc.language.iso.fl_str_mv por
language por
dc.rights.driver.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.source.none.fl_str_mv reponame:Repositório Institucional da Universidade Federal do Ceará (UFC)
instname:Universidade Federal do Ceará (UFC)
instacron:UFC
instname_str Universidade Federal do Ceará (UFC)
instacron_str UFC
institution UFC
reponame_str Repositório Institucional da Universidade Federal do Ceará (UFC)
collection Repositório Institucional da Universidade Federal do Ceará (UFC)
bitstream.url.fl_str_mv http://repositorio.ufc.br/bitstream/riufc/81643/3/2025_dis_jmsantosjunior.pdf
http://repositorio.ufc.br/bitstream/riufc/81643/4/license.txt
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repository.mail.fl_str_mv bu@ufc.br || repositorio@ufc.br
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