A Frostman type lemma for sets with large intersections, and an application to Diophantine approximation

We consider classes \(\mathscr{G}^s ([0,1])\) of subsets of \([0,1]\), originally introduced by Falconer, that are closed under countable intersections, and such that every set in the class has Hausdorff dimension at least \(s\). We provide a Frostman type lemma to determine if a limsup-set is in su...

Full description

Saved in:
Bibliographic Details
Published inarXiv.org
Main Authors Persson, Tomas, Reeve, Henry W J
Format Paper Journal Article
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 11.09.2017
Subjects
Online AccessGet full text

Cover

Loading…
Abstract We consider classes \(\mathscr{G}^s ([0,1])\) of subsets of \([0,1]\), originally introduced by Falconer, that are closed under countable intersections, and such that every set in the class has Hausdorff dimension at least \(s\). We provide a Frostman type lemma to determine if a limsup-set is in such a class. Suppose \(E = \limsup E_n \subset [0,1]\), and that \(\mu_n\) are probability measures with support in \(E_n\). If there is a constant \(C\) such that \[\iint|x-y|^{-s}\, \mathrm{d}\mu_n(x)\mathrm{d}\mu_n(y)<C\] for all \(n\), then under suitable conditions on the limit measure of the sequence \((\mu_n)\), we prove that the set \(E\) is in the class \(\mathscr{G}^s ([0,1])\). As an application we prove that for \(\alpha > 1\) and almost all \(\lambda \in (\frac{1}{2},1)\) the set \[ E_\lambda(\alpha) = \{\,x\in[0,1] : |x - s_n| < 2^{-\alpha n} \text{infinitely often}\ \}\] where \(s_n \in \{\,(1-\lambda)\sum_{k=0}^na_k\lambda^k\) and \(a_k\in\{0,1\}\,\}\), belongs to the class \(\mathscr{G}^s\) for \(s \leq \frac{1}{\alpha}\). This improves one of our previous results.
AbstractList Proceedings of the Edinburgh Mathematical Society, Volume 58, Issue 02, June 2015, 521--542 We consider classes $\mathscr{G}^s ([0,1])$ of subsets of $[0,1]$, originally introduced by Falconer, that are closed under countable intersections, and such that every set in the class has Hausdorff dimension at least $s$. We provide a Frostman type lemma to determine if a limsup-set is in such a class. Suppose $E = \limsup E_n \subset [0,1]$, and that $\mu_n$ are probability measures with support in $E_n$. If there is a constant $C$ such that \[\iint|x-y|^{-s}\, \mathrm{d}\mu_n(x)\mathrm{d}\mu_n(y)<C\] for all $n$, then under suitable conditions on the limit measure of the sequence $(\mu_n)$, we prove that the set $E$ is in the class $\mathscr{G}^s ([0,1])$. As an application we prove that for $\alpha > 1$ and almost all $\lambda \in (\frac{1}{2},1)$ the set \[ E_\lambda(\alpha) = \{\,x\in[0,1] : |x - s_n| < 2^{-\alpha n} \text{infinitely often}\ \}\] where $s_n \in \{\,(1-\lambda)\sum_{k=0}^na_k\lambda^k$ and $a_k\in\{0,1\}\,\}$, belongs to the class $\mathscr{G}^s$ for $s \leq \frac{1}{\alpha}$. This improves one of our previous results.
We consider classes \(\mathscr{G}^s ([0,1])\) of subsets of \([0,1]\), originally introduced by Falconer, that are closed under countable intersections, and such that every set in the class has Hausdorff dimension at least \(s\). We provide a Frostman type lemma to determine if a limsup-set is in such a class. Suppose \(E = \limsup E_n \subset [0,1]\), and that \(\mu_n\) are probability measures with support in \(E_n\). If there is a constant \(C\) such that \[\iint|x-y|^{-s}\, \mathrm{d}\mu_n(x)\mathrm{d}\mu_n(y)<C\] for all \(n\), then under suitable conditions on the limit measure of the sequence \((\mu_n)\), we prove that the set \(E\) is in the class \(\mathscr{G}^s ([0,1])\). As an application we prove that for \(\alpha > 1\) and almost all \(\lambda \in (\frac{1}{2},1)\) the set \[ E_\lambda(\alpha) = \{\,x\in[0,1] : |x - s_n| < 2^{-\alpha n} \text{infinitely often}\ \}\] where \(s_n \in \{\,(1-\lambda)\sum_{k=0}^na_k\lambda^k\) and \(a_k\in\{0,1\}\,\}\), belongs to the class \(\mathscr{G}^s\) for \(s \leq \frac{1}{\alpha}\). This improves one of our previous results.
Author Persson, Tomas
Reeve, Henry W J
Author_xml – sequence: 1
  givenname: Tomas
  surname: Persson
  fullname: Persson, Tomas
– sequence: 2
  givenname: Henry
  surname: Reeve
  middlename: W J
  fullname: Reeve, Henry W J
BackLink https://doi.org/10.1017/S0013091514000066$$DView published paper (Access to full text may be restricted)
https://doi.org/10.48550/arXiv.1302.0954$$DView paper in arXiv
BookMark eNotkM1LAzEQxYMoWGvvniTg1a35TvdYqlWh4KX3ZXabtSm7yZqk2v733W09DA_mPWYevzt07bwzCD1QMhUzKckLhIP9nVJO2JTkUlyhEeOcZjPB2C2axLgjhDClmZR8hLZzvAw-phYcTsfO4Ma0LeDaBxxNivjPpi1uIHwbbF0yIZoqWe_iMwa36QdD1zW2gmGJk8ev1ndbcMk6M1jBH2x7Nu_RTQ1NNJN_HaP18m29-MhWX--fi_kqA0lVJjbaQG54TQnXSlSMm1rqihmSC1FqwsoZBcpAK5rnQgtRAy8JqwhRmlNZ8jF6vJw9Uyi60L8Px2KgUQw0-sDTJdB3-9mbmIqd3wfXVyoY0YoLpaXiJ0_GZDI
ContentType Paper
Journal Article
Copyright 2017. This work is published under http://arxiv.org/licenses/nonexclusive-distrib/1.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.
http://arxiv.org/licenses/nonexclusive-distrib/1.0
Copyright_xml – notice: 2017. This work is published under http://arxiv.org/licenses/nonexclusive-distrib/1.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.
– notice: http://arxiv.org/licenses/nonexclusive-distrib/1.0
DBID 8FE
8FG
ABJCF
ABUWG
AFKRA
AZQEC
BENPR
BGLVJ
CCPQU
DWQXO
HCIFZ
L6V
M7S
PIMPY
PQEST
PQQKQ
PQUKI
PRINS
PTHSS
AKZ
GOX
DOI 10.48550/arxiv.1302.0954
DatabaseName ProQuest SciTech Collection
ProQuest Technology Collection
Materials Science & Engineering Collection
ProQuest Central (Alumni)
ProQuest Central
ProQuest Central Essentials
ProQuest Central
Technology Collection
ProQuest One Community College
ProQuest Central Korea
SciTech Premium Collection
ProQuest Engineering Collection
Engineering Database
Publicly Available Content Database
ProQuest One Academic Eastern Edition (DO NOT USE)
ProQuest One Academic
ProQuest One Academic UKI Edition
ProQuest Central China
Engineering Collection
arXiv Mathematics
arXiv.org
DatabaseTitle Publicly Available Content Database
Engineering Database
Technology Collection
ProQuest Central Essentials
ProQuest One Academic Eastern Edition
ProQuest Central (Alumni Edition)
SciTech Premium Collection
ProQuest One Community College
ProQuest Technology Collection
ProQuest SciTech Collection
ProQuest Central China
ProQuest Central
ProQuest Engineering Collection
ProQuest One Academic UKI Edition
ProQuest Central Korea
Materials Science & Engineering Collection
ProQuest One Academic
Engineering Collection
DatabaseTitleList
Publicly Available Content Database
Database_xml – sequence: 1
  dbid: GOX
  name: arXiv.org
  url: http://arxiv.org/find
  sourceTypes: Open Access Repository
– sequence: 2
  dbid: 8FG
  name: ProQuest Technology Collection
  url: https://search.proquest.com/technologycollection1
  sourceTypes: Aggregation Database
DeliveryMethod fulltext_linktorsrc
Discipline Physics
EISSN 2331-8422
ExternalDocumentID 1302_0954
Genre Working Paper/Pre-Print
GroupedDBID 8FE
8FG
ABJCF
ABUWG
AFKRA
ALMA_UNASSIGNED_HOLDINGS
AZQEC
BENPR
BGLVJ
CCPQU
DWQXO
FRJ
HCIFZ
L6V
M7S
M~E
PIMPY
PQEST
PQQKQ
PQUKI
PRINS
PTHSS
AKZ
GOX
ID FETCH-LOGICAL-a516-4d7ea9e3f103764c23ef57c2e0944b702b81a12a761994744fa3b02c0067315b3
IEDL.DBID GOX
IngestDate Mon Jan 08 05:45:01 EST 2024
Thu Oct 10 18:45:17 EDT 2024
IsDoiOpenAccess true
IsOpenAccess true
IsPeerReviewed false
IsScholarly false
Language English
LinkModel DirectLink
MergedId FETCHMERGED-LOGICAL-a516-4d7ea9e3f103764c23ef57c2e0944b702b81a12a761994744fa3b02c0067315b3
OpenAccessLink https://arxiv.org/abs/1302.0954
PQID 2076346756
PQPubID 2050157
ParticipantIDs arxiv_primary_1302_0954
proquest_journals_2076346756
PublicationCentury 2000
PublicationDate 20170911
PublicationDateYYYYMMDD 2017-09-11
PublicationDate_xml – month: 09
  year: 2017
  text: 20170911
  day: 11
PublicationDecade 2010
PublicationPlace Ithaca
PublicationPlace_xml – name: Ithaca
PublicationTitle arXiv.org
PublicationYear 2017
Publisher Cornell University Library, arXiv.org
Publisher_xml – name: Cornell University Library, arXiv.org
SSID ssj0002672553
Score 1.6796205
SecondaryResourceType preprint
Snippet We consider classes \(\mathscr{G}^s ([0,1])\) of subsets of \([0,1]\), originally introduced by Falconer, that are closed under countable intersections, and...
Proceedings of the Edinburgh Mathematical Society, Volume 58, Issue 02, June 2015, 521--542 We consider classes $\mathscr{G}^s ([0,1])$ of subsets of $[0,1]$,...
SourceID arxiv
proquest
SourceType Open Access Repository
Aggregation Database
SubjectTerms Intersections
Mathematics - Dynamical Systems
Mathematics - Number Theory
SummonAdditionalLinks – databaseName: ProQuest Technology Collection
  dbid: 8FG
  link: http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwfV3PS8MwFA66IXjzt9MpOXg0uqRp2p5E1DkExcOE3UqSJmywtbWtsj_fvKxTQfDQS0MvL817X77340PoQjAr3BInQgWW8EQmREnKiNSGDUIWU6Mgo_v8IkZv_GkSTlrCrW7LKtc-0TvqrNDAkQMTIgJ3qkNxU74TUI2C7GorobGJuhQm4UGn-PDxm2NhInKIOVhlJ_3ormtZLWefIIHMrhy44A6S-jd_PLEPL8Md1H2Vpal20YbJ99CWr8rU9T6a3mLflbGQOQauFM_NYiGxw5m4Nk2NgUTFc6jlxjD2oap9XVVeX2KZZ-7Bv9LTuCnw_awopxK0IQz2w8SXs1Xn4gEaDx_GdyPSSiMQGVJBeBYZmZjAQpef4JoFxoaRZsZd1riKBkzF1FlcAkeR8IhzKwM1YNrL0tBQBYeokxe5OUbY4QVrqXS3Qvepi_cytg4VMZspHSdaJz105C2UlqvpF5D_YinYrof6a5ul7Y9fpz_bdPL_8inaZhAhQY2B9lGnqT7MmYvvjTr3m_gFhkqlhw
  priority: 102
  providerName: ProQuest
Title A Frostman type lemma for sets with large intersections, and an application to Diophantine approximation
URI https://www.proquest.com/docview/2076346756
https://arxiv.org/abs/1302.0954
hasFullText 1
inHoldings 1
isFullTextHit
isPrint
link http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwdV07T8MwED61ZWFBIF6FUjwwEogd5zUW6ENILQgVqVtkp7ZaqU2rJqBO_HbOToqQEEMyJPZytu713X0HcBMwHeAv7gTS0w6PRexIQZkjUsVcn0VUSYPoDkfB4J0_T_xJDa53vTBis51_lvzAMr83qNodOgG8DnXGTMVW_2VSgo2Wiata_rMMPUz75Y9itdaidwgHlZtHOuW5HEFNZccw6xDbZLEUGTGpT7JQy6Ug6DaSXBU5MTlRsjCl2cSwOGxyWyaV5bcEw318yC-0mRQr8jRfrWfCjHpQxHKDb-dlI-IJjHvd8ePAqSYdOMKngcOnoRKx8rRp2gt4yjyl_TBlCmMvLkOXyYiiAIVJOcQ85FwLT7ostVNmqC-9U2hkq0ydA0HzrzUVGOThVjTfItLo5DA9lWkUp2nchDMroWRdklkYOIslRnZNaO1kllT3OE-Yi_oHdakfXPy78RL2mbF1Zq4CbUGj2HyoK7TUhWxDPer127D30B29vrXt6eF7-NX9BsGpmCA
link.rule.ids 228,230,786,790,891,12792,21416,27956,33406,33777,43633,43838
linkProvider Cornell University
linkToHtml http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwfV3NT8IwFG8UYvTmtyhqDx6dsq7rtpPxi6ACIQYTbktb2kAC29ym4c-3rww1MfGwy5pdXtf3fv29jx9CF4xoZpaow4SnHRrxyBHcJQ6XirR8ErpKQEa312edN_o88kcV4VZUZZUrn2gd9TiVwJEDE8I8c6p9dpO9O6AaBdnVSkJjHdVh5GZYQ_W7x_7g9ZtlISwwmNlb5ift8K5rni-mnyCCTK4MvKAGlNo3f3yxDTDtbVQf8EzlO2hNJbtow9ZlymIPTW6x7cuY8wQDW4pnaj7n2CBNXKiywECj4hlUc2MY_JAXtrIqKS4xT8bmwb8S1LhM8cM0zSYc1CEUtuPEF9Nl7-I-GrYfh_cdpxJHcLjvMoeOA8Uj5Wno82NUEk9pP5BEmesaFUGLiNA1NufAUkQ0oFRzT7SItMI0ri-8A1RL0kQdIWwQg9YuN_dC86mJ-DzUBhcRPRYyjKSMGujQWijOlvMvIANGYrBdAzVXNourX7-Ifzbq-P_lc7TZGfa6cfep_3KCtgjES9BmcJuoVuYf6tRE-1KcVVv6BVpIqd4
openUrl ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=A+Frostman+type+lemma+for+sets+with+large+intersections%2C+and+an+application+to+Diophantine+approximation&rft.jtitle=arXiv.org&rft.au=Persson%2C+Tomas&rft.au=Reeve%2C+Henry+W+J&rft.date=2017-09-11&rft.pub=Cornell+University+Library%2C+arXiv.org&rft.eissn=2331-8422&rft_id=info:doi/10.48550%2Farxiv.1302.0954