New bounds on the dimensions of planar distance sets

Autores
Keleti, Tamas; Shmerkin, Pablo Sebastian
Año de publicación
2019
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
We prove new bounds on the dimensions of distance sets and pinned distance sets of planar sets. Among other results, we show that if A ⊂ R2 is a Borel set of Hausdorff dimension s > 1, then its distance set has Hausdorff dimension at least 37/54 ≈ 0.685. Moreover, if s ∈ (1, 3/2], then outside of a set of exceptional y of Hausdorff dimension at most 1, the pinned distance set {|x − y| : x ∈ A} has Hausdorff dimension ≥ 2 3 s and packing dimension at least 1 4 (1+s+3s(2 − s)) ≥ 0.933. These estimates improve upon the existing ones by Bourgain, Wolff, Peres–Schlag and Iosevich–Liu for sets of Hausdorff dimension > 1. Our proof uses a multi-scale decomposition of measures in which, unlike previous works, we are able to choose the scales subject to certain constrains. This leads to a combinatorial problem, which is a key new ingredient of our approach, and which we solve completely by optimizing certain variation of Lipschitz functions.
Fil: Keleti, Tamas. Eötvös University; Argentina
Fil: Shmerkin, Pablo Sebastian. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Materia
DISTANCE SETS
FALCONER’S PROBLEM
HAUSDORFF DIMENSION
LIPSCHITZ FUNCTIONS
PACKING DIMENSION
PINNED DISTANCE SETS
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/135439
Acceder
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oai_identifier_str oai:ri.conicet.gov.ar:11336/135439
network_acronym_str CONICETDig
repository_id_str 3498
network_name_str CONICET Digital (CONICET)
spelling New bounds on the dimensions of planar distance setsKeleti, TamasShmerkin, Pablo SebastianDISTANCE SETSFALCONER’S PROBLEMHAUSDORFF DIMENSIONLIPSCHITZ FUNCTIONSPACKING DIMENSIONPINNED DISTANCE SETShttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1We prove new bounds on the dimensions of distance sets and pinned distance sets of planar sets. Among other results, we show that if A ⊂ R2 is a Borel set of Hausdorff dimension s > 1, then its distance set has Hausdorff dimension at least 37/54 ≈ 0.685. Moreover, if s ∈ (1, 3/2], then outside of a set of exceptional y of Hausdorff dimension at most 1, the pinned distance set {|x − y| : x ∈ A} has Hausdorff dimension ≥ 2 3 s and packing dimension at least 1 4 (1+s+3s(2 − s)) ≥ 0.933. These estimates improve upon the existing ones by Bourgain, Wolff, Peres–Schlag and Iosevich–Liu for sets of Hausdorff dimension > 1. Our proof uses a multi-scale decomposition of measures in which, unlike previous works, we are able to choose the scales subject to certain constrains. This leads to a combinatorial problem, which is a key new ingredient of our approach, and which we solve completely by optimizing certain variation of Lipschitz functions.Fil: Keleti, Tamas. Eötvös University; ArgentinaFil: Shmerkin, Pablo Sebastian. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaBirkhauser Verlag Ag2019-07info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/135439Keleti, Tamas; Shmerkin, Pablo Sebastian; New bounds on the dimensions of planar distance sets; Birkhauser Verlag Ag; Geometric and Functional Analysis; 29; 6; 7-2019; 1886-19481016-443X1420-8970CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/doi/10.1007/s00039-019-00500-9info:eu-repo/semantics/altIdentifier/url/https://link.springer.com/article/10.1007/s00039-019-00500-9info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1801.08745info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-03T09:53:28Zoai:ri.conicet.gov.ar:11336/135439instacron: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:34982025-09-03 09:53:29.241CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv New bounds on the dimensions of planar distance sets
title New bounds on the dimensions of planar distance sets
spellingShingle New bounds on the dimensions of planar distance sets
Keleti, Tamas
DISTANCE SETS
FALCONER’S PROBLEM
HAUSDORFF DIMENSION
LIPSCHITZ FUNCTIONS
PACKING DIMENSION
PINNED DISTANCE SETS
title_short New bounds on the dimensions of planar distance sets
title_full New bounds on the dimensions of planar distance sets
title_fullStr New bounds on the dimensions of planar distance sets
title_full_unstemmed New bounds on the dimensions of planar distance sets
title_sort New bounds on the dimensions of planar distance sets
dc.creator.none.fl_str_mv Keleti, Tamas
Shmerkin, Pablo Sebastian
author Keleti, Tamas
author_facet Keleti, Tamas
Shmerkin, Pablo Sebastian
author_role author
author2 Shmerkin, Pablo Sebastian
author2_role author
dc.subject.none.fl_str_mv DISTANCE SETS
FALCONER’S PROBLEM
HAUSDORFF DIMENSION
LIPSCHITZ FUNCTIONS
PACKING DIMENSION
PINNED DISTANCE SETS
topic DISTANCE SETS
FALCONER’S PROBLEM
HAUSDORFF DIMENSION
LIPSCHITZ FUNCTIONS
PACKING DIMENSION
PINNED DISTANCE SETS
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv We prove new bounds on the dimensions of distance sets and pinned distance sets of planar sets. Among other results, we show that if A ⊂ R2 is a Borel set of Hausdorff dimension s > 1, then its distance set has Hausdorff dimension at least 37/54 ≈ 0.685. Moreover, if s ∈ (1, 3/2], then outside of a set of exceptional y of Hausdorff dimension at most 1, the pinned distance set {|x − y| : x ∈ A} has Hausdorff dimension ≥ 2 3 s and packing dimension at least 1 4 (1+s+3s(2 − s)) ≥ 0.933. These estimates improve upon the existing ones by Bourgain, Wolff, Peres–Schlag and Iosevich–Liu for sets of Hausdorff dimension > 1. Our proof uses a multi-scale decomposition of measures in which, unlike previous works, we are able to choose the scales subject to certain constrains. This leads to a combinatorial problem, which is a key new ingredient of our approach, and which we solve completely by optimizing certain variation of Lipschitz functions.
Fil: Keleti, Tamas. Eötvös University; Argentina
Fil: Shmerkin, Pablo Sebastian. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
description We prove new bounds on the dimensions of distance sets and pinned distance sets of planar sets. Among other results, we show that if A ⊂ R2 is a Borel set of Hausdorff dimension s > 1, then its distance set has Hausdorff dimension at least 37/54 ≈ 0.685. Moreover, if s ∈ (1, 3/2], then outside of a set of exceptional y of Hausdorff dimension at most 1, the pinned distance set {|x − y| : x ∈ A} has Hausdorff dimension ≥ 2 3 s and packing dimension at least 1 4 (1+s+3s(2 − s)) ≥ 0.933. These estimates improve upon the existing ones by Bourgain, Wolff, Peres–Schlag and Iosevich–Liu for sets of Hausdorff dimension > 1. Our proof uses a multi-scale decomposition of measures in which, unlike previous works, we are able to choose the scales subject to certain constrains. This leads to a combinatorial problem, which is a key new ingredient of our approach, and which we solve completely by optimizing certain variation of Lipschitz functions.
publishDate 2019
dc.date.none.fl_str_mv 2019-07
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/11336/135439
Keleti, Tamas; Shmerkin, Pablo Sebastian; New bounds on the dimensions of planar distance sets; Birkhauser Verlag Ag; Geometric and Functional Analysis; 29; 6; 7-2019; 1886-1948
1016-443X
1420-8970
CONICET Digital
CONICET
url http://hdl.handle.net/11336/135439
identifier_str_mv Keleti, Tamas; Shmerkin, Pablo Sebastian; New bounds on the dimensions of planar distance sets; Birkhauser Verlag Ag; Geometric and Functional Analysis; 29; 6; 7-2019; 1886-1948
1016-443X
1420-8970
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/doi/10.1007/s00039-019-00500-9
info:eu-repo/semantics/altIdentifier/url/https://link.springer.com/article/10.1007/s00039-019-00500-9
info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1801.08745
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
application/pdf
dc.publisher.none.fl_str_mv Birkhauser Verlag Ag
publisher.none.fl_str_mv Birkhauser Verlag Ag
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
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 13.13397

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