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dc.contributor.authorDąbrowski, Damian
dc.contributor.authorOrponen, Tuomas
dc.contributor.authorWang, Hong
dc.date.accessioned2024-05-03T08:31:31Z
dc.date.available2024-05-03T08:31:31Z
dc.date.issued2024
dc.identifier.citationDąbrowski, D., Orponen, T., & Wang, H. (2024). How much can heavy lines cover?. <i>Journal of the London Mathematical Society</i>, <i>109</i>(5), Article e12910. <a href="https://doi.org/10.1112/jlms.12910" target="_blank">https://doi.org/10.1112/jlms.12910</a>
dc.identifier.otherCONVID_213478190
dc.identifier.urihttps://jyx.jyu.fi/handle/123456789/94677
dc.description.abstractOne formulation of Marstrand’s slicing theorem is the following. Assume that𝑡∈(1,2],and𝐵⊂ℝ2is a Borel set with 𝑡(𝐵)<∞. Then, for almost all directions𝑒∈𝑆1, 𝑡 almost all of 𝐵 is covered by lines𝓁parallel to 𝑒 with dim H(𝐵∩𝓁)=𝑡−1. We investigate the prospects of sharpening Marstrand’s result in the following sense: in a generic direction𝑒∈𝑆1, is it true that a strictly less than 𝑡-dimensional part of𝐵is covered by the heavy lines𝓁⊂ℝ2, namely those with dim H(𝐵∩𝓁)>𝑡−1? A positive answer for𝑡-regular sets𝐵⊂ℝ2was previously obtained by the first author. The answer for general Borel sets turns out to be negative for𝑡∈(1,32] and positive for𝑡∈(32,2]. More precisely, the heavy lines can cover up to amin{𝑡,3−𝑡} dimensional part of𝐵in a generic direction. We also consider the part of𝐵covered by the𝑠-heavy lines, namely those with dim H(𝐵∩𝓁)⩾𝑠for𝑠>𝑡−1. We establish a sharp answer to the question: how much can the𝑠-heavy lines cover in a generic direction? Finally, we identify a new class of sets called sub-uniformly distributed sets, which generalise Ahlfors-regular sets. Roughly speaking, these sets share the spatial uniformity of Ahlfors-regular sets, but pose no restrictions on uniformity across different scales. We then extend and sharpen the first author’s previous result on Ahlfors-regular sets to the class of sub uniformly distributed sets.en
dc.format.mimetypeapplication/pdf
dc.language.isoeng
dc.publisherWiley-Blackwell
dc.relation.ispartofseriesJournal of the London Mathematical Society
dc.rightsCC BY 4.0
dc.titleHow much can heavy lines cover?
dc.typearticle
dc.identifier.urnURN:NBN:fi:jyu-202405033304
dc.contributor.laitosMatematiikan ja tilastotieteen laitosfi
dc.contributor.laitosDepartment of Mathematics and Statisticsen
dc.type.urihttp://purl.org/eprint/type/JournalArticle
dc.type.coarhttp://purl.org/coar/resource_type/c_2df8fbb1
dc.description.reviewstatuspeerReviewed
dc.relation.issn0024-6107
dc.relation.numberinseries5
dc.relation.volume109
dc.type.versionpublishedVersion
dc.rights.copyright© 2024 The Authors.Journal of the London Mathematical Society is copyright © London Mathematical Society
dc.rights.accesslevelopenAccessfi
dc.relation.grantnumber347123
dc.relation.grantnumber101087499
dc.relation.grantnumber101087499
dc.relation.grantnumber355453
dc.relation.projectidinfo:eu-repo/grantAgreement/EC/H2020/101087499/EU//MUSING
dc.format.contentfulltext
dc.rights.urlhttps://creativecommons.org/licenses/by/4.0/
dc.relation.doi10.1112/jlms.12910
dc.relation.funderResearch Council of Finlanden
dc.relation.funderEuropean Commissionen
dc.relation.funderResearch Council of Finlanden
dc.relation.funderSuomen Akatemiafi
dc.relation.funderEuroopan komissiofi
dc.relation.funderSuomen Akatemiafi
jyx.fundingprogramPostdoctoral Researcher, AoFen
jyx.fundingprogramERC Consolidator Grant, HEen
jyx.fundingprogramAcademy Project, AoFen
jyx.fundingprogramTutkijatohtori, SAfi
jyx.fundingprogramERC Consolidator Grant, HEfi
jyx.fundingprogramAkatemiahanke, SAfi
jyx.fundinginformationResearch Council of Finland, Grant/Award Numbers: 347123, 355453; European Research Council, Grant/Award Number: 101087499; NSF, Grant/Award Numbers: DMS-2238818, DMS-2055544.
dc.type.okmA1


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