Salem Sets with No Arithmetic Progressions

We construct compact Salem sets in R/Z of any dimension (including 1), which do not contain any arithmetic progressions of length 3. Moreover, the sets can be taken to be Ahlfors regular if the dimension is less than 1, and the measure witnessing the Fourier decay can be taken to be Frostman in the...

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Detalles Bibliográficos
Autor: Shmerkin, Pablo Sebastian
Tipo de recurso: artículo
Estado:Versión publicada
Fecha de publicación:2017
País:Argentina
Institución:Consejo Nacional de Investigaciones Científicas y Técnicas
Repositorio:CONICET Digital (CONICET)
Idioma:inglés
OAI Identifier:oai:ri.conicet.gov.ar:11336/72608
Acceso en línea:http://hdl.handle.net/11336/72608
Access Level:acceso abierto
Palabra clave:ARITHMETIC PROGRESSIONS
SALEM SETS
PSEUDO-RANDOMNESS
https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
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repository_id_str
spelling Salem Sets with No Arithmetic ProgressionsShmerkin, Pablo SebastianARITHMETIC PROGRESSIONSSALEM SETSPSEUDO-RANDOMNESShttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1We construct compact Salem sets in R/Z of any dimension (including 1), which do not contain any arithmetic progressions of length 3. Moreover, the sets can be taken to be Ahlfors regular if the dimension is less than 1, and the measure witnessing the Fourier decay can be taken to be Frostman in the case of dimension 1. This is in sharp contrast to the situation in the discrete setting (where Fourier uniformity is well known to imply existence of progressions) and helps clarify a result of Łaba and Pramanik on pseudo-random subsets of R which do contain progressions.Fil: Shmerkin, Pablo Sebastian. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaOxford University Press2017-04info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/72608Shmerkin, Pablo Sebastian; Salem Sets with No Arithmetic Progressions; Oxford University Press; International Mathematics Research Notices; 2017; 7; 4-2017; 1929-19411073-7928CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1510.07596info:eu-repo/semantics/altIdentifier/doi/10.1093/imrn/rnw097info:eu-repo/semantics/altIdentifier/url/https://academic.oup.com/imrn/article-abstract/2017/7/1929/3060565info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2024-05-08T13:32:16Zoai:ri.conicet.gov.ar:11336/72608instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982024-05-08 13:32:16.4CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Salem Sets with No Arithmetic Progressions
title Salem Sets with No Arithmetic Progressions
spellingShingle Salem Sets with No Arithmetic Progressions
Shmerkin, Pablo Sebastian
ARITHMETIC PROGRESSIONS
SALEM SETS
PSEUDO-RANDOMNESS
https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
title_short Salem Sets with No Arithmetic Progressions
title_full Salem Sets with No Arithmetic Progressions
title_fullStr Salem Sets with No Arithmetic Progressions
title_full_unstemmed Salem Sets with No Arithmetic Progressions
title_sort Salem Sets with No Arithmetic Progressions
dc.creator.none.fl_str_mv Shmerkin, Pablo Sebastian
author Shmerkin, Pablo Sebastian
author_facet Shmerkin, Pablo Sebastian
author_role author
dc.subject.none.fl_str_mv ARITHMETIC PROGRESSIONS
SALEM SETS
PSEUDO-RANDOMNESS
https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
topic ARITHMETIC PROGRESSIONS
SALEM SETS
PSEUDO-RANDOMNESS
https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
description We construct compact Salem sets in R/Z of any dimension (including 1), which do not contain any arithmetic progressions of length 3. Moreover, the sets can be taken to be Ahlfors regular if the dimension is less than 1, and the measure witnessing the Fourier decay can be taken to be Frostman in the case of dimension 1. This is in sharp contrast to the situation in the discrete setting (where Fourier uniformity is well known to imply existence of progressions) and helps clarify a result of Łaba and Pramanik on pseudo-random subsets of R which do contain progressions.
publishDate 2017
dc.date.none.fl_str_mv 2017-04
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/72608
Shmerkin, Pablo Sebastian; Salem Sets with No Arithmetic Progressions; Oxford University Press; International Mathematics Research Notices; 2017; 7; 4-2017; 1929-1941
1073-7928
CONICET Digital
CONICET
url http://hdl.handle.net/11336/72608
identifier_str_mv Shmerkin, Pablo Sebastian; Salem Sets with No Arithmetic Progressions; Oxford University Press; International Mathematics Research Notices; 2017; 7; 4-2017; 1929-1941
1073-7928
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1510.07596
info:eu-repo/semantics/altIdentifier/doi/10.1093/imrn/rnw097
info:eu-repo/semantics/altIdentifier/url/https://academic.oup.com/imrn/article-abstract/2017/7/1929/3060565
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv Oxford University Press
publisher.none.fl_str_mv Oxford University Press
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
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score 15.811543

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