On distance sets, box-counting and Ahlfors regular sets

Autores
Shmerkin, Pablo Sebastian
Año de publicación
2017
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
We obtain box-counting estimates for the pinned distance sets of (dense subsets of) planar discrete Ahlfors-regular sets of exponent s > 1. As a corollary, we improve upon a recent result of Orponen, by showing that if A is Ahlfors-regular of dimension s > 1, then almost all pinned distance sets of A have lower box-counting dimension 1. We also show that if A,B ⊂ R 2 have Hausdorff dimension greater than 1 and A is Ahlfors-regular, then the set of distances between A and B has modified lower box-counting dimension 1, which taking B = A improves Orponen’s result in a different direction, by lowering packing dimension to modified lower box-counting dimension. The proofs involve ergodic-theoretic ideas, relying on the theory of CP-processes and projections.
Fil: Shmerkin, Pablo Sebastian. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentina
Materia
Distance sets
Box dimension
Ahlfors-regular sets
CP-processes
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:11336/77253
Acceder
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spelling On distance sets, box-counting and Ahlfors regular setsShmerkin, Pablo SebastianDistance setsBox dimensionAhlfors-regular setsCP-processeshttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1We obtain box-counting estimates for the pinned distance sets of (dense subsets of) planar discrete Ahlfors-regular sets of exponent s > 1. As a corollary, we improve upon a recent result of Orponen, by showing that if A is Ahlfors-regular of dimension s > 1, then almost all pinned distance sets of A have lower box-counting dimension 1. We also show that if A,B ⊂ R 2 have Hausdorff dimension greater than 1 and A is Ahlfors-regular, then the set of distances between A and B has modified lower box-counting dimension 1, which taking B = A improves Orponen’s result in a different direction, by lowering packing dimension to modified lower box-counting dimension. The proofs involve ergodic-theoretic ideas, relying on the theory of CP-processes and projections.Fil: Shmerkin, Pablo Sebastian. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; ArgentinaAlliance of Diamond Open Access Journals2017-05info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/77253Shmerkin, Pablo Sebastian; On distance sets, box-counting and Ahlfors regular sets; Alliance of Diamond Open Access Journals; Discrete Analysis; 2017; 9; 5-2017; 1-222397-3129CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://discreteanalysisjournal.com/article/1643-on-distance-sets-box-counting-and-ahlfors-regular-setinfo:eu-repo/semantics/altIdentifier/doi/10.19086/da.1643info:eu-repo/semantics/altIdentifier/arxiv/arXiv:1605.00187info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-03T09:50:31Zoai:ri.conicet.gov.ar:11336/77253instacron: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 09:50:31.368CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv On distance sets, box-counting and Ahlfors regular sets
title On distance sets, box-counting and Ahlfors regular sets
spellingShingle On distance sets, box-counting and Ahlfors regular sets
Shmerkin, Pablo Sebastian
Distance sets
Box dimension
Ahlfors-regular sets
CP-processes
title_short On distance sets, box-counting and Ahlfors regular sets
title_full On distance sets, box-counting and Ahlfors regular sets
title_fullStr On distance sets, box-counting and Ahlfors regular sets
title_full_unstemmed On distance sets, box-counting and Ahlfors regular sets
title_sort On distance sets, box-counting and Ahlfors regular sets
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 Distance sets
Box dimension
Ahlfors-regular sets
CP-processes
topic Distance sets
Box dimension
Ahlfors-regular sets
CP-processes
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 obtain box-counting estimates for the pinned distance sets of (dense subsets of) planar discrete Ahlfors-regular sets of exponent s > 1. As a corollary, we improve upon a recent result of Orponen, by showing that if A is Ahlfors-regular of dimension s > 1, then almost all pinned distance sets of A have lower box-counting dimension 1. We also show that if A,B ⊂ R 2 have Hausdorff dimension greater than 1 and A is Ahlfors-regular, then the set of distances between A and B has modified lower box-counting dimension 1, which taking B = A improves Orponen’s result in a different direction, by lowering packing dimension to modified lower box-counting dimension. The proofs involve ergodic-theoretic ideas, relying on the theory of CP-processes and projections.
Fil: Shmerkin, Pablo Sebastian. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentina
description We obtain box-counting estimates for the pinned distance sets of (dense subsets of) planar discrete Ahlfors-regular sets of exponent s > 1. As a corollary, we improve upon a recent result of Orponen, by showing that if A is Ahlfors-regular of dimension s > 1, then almost all pinned distance sets of A have lower box-counting dimension 1. We also show that if A,B ⊂ R 2 have Hausdorff dimension greater than 1 and A is Ahlfors-regular, then the set of distances between A and B has modified lower box-counting dimension 1, which taking B = A improves Orponen’s result in a different direction, by lowering packing dimension to modified lower box-counting dimension. The proofs involve ergodic-theoretic ideas, relying on the theory of CP-processes and projections.
publishDate 2017
dc.date.none.fl_str_mv 2017-05
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/11336/77253
Shmerkin, Pablo Sebastian; On distance sets, box-counting and Ahlfors regular sets; Alliance of Diamond Open Access Journals; Discrete Analysis; 2017; 9; 5-2017; 1-22
2397-3129
CONICET Digital
CONICET
url http://hdl.handle.net/11336/77253
identifier_str_mv Shmerkin, Pablo Sebastian; On distance sets, box-counting and Ahlfors regular sets; Alliance of Diamond Open Access Journals; Discrete Analysis; 2017; 9; 5-2017; 1-22
2397-3129
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://discreteanalysisjournal.com/article/1643-on-distance-sets-box-counting-and-ahlfors-regular-set
info:eu-repo/semantics/altIdentifier/doi/10.19086/da.1643
info:eu-repo/semantics/altIdentifier/arxiv/arXiv:1605.00187
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 Alliance of Diamond Open Access Journals
publisher.none.fl_str_mv Alliance of Diamond Open Access Journals
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
_version_ 1842269035812618240
score 13.13397

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