{"id":357527,"date":"2017-01-25T10:46:51","date_gmt":"2017-01-25T18:46:51","guid":{"rendered":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/?post_type=msr-research-item&p=357527"},"modified":"2018-10-16T19:59:52","modified_gmt":"2018-10-17T02:59:52","slug":"dimensions-fractals-defined-via-semigroup-generated-2-3","status":"publish","type":"msr-research-item","link":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/publication\/dimensions-fractals-defined-via-semigroup-generated-2-3\/","title":{"rendered":"Dimensions Of Some Fractals Defined Via The Semigroup Generated By 2 And 3"},"content":{"rendered":"

We compute the Hausdorff and Minkowski dimension of subsets of the symbolic space $\\Sigma_m=\\{0,…,m-1\\}^\\N$<\/span><\/span><\/span> that are invariant under multiplication by integers. The results apply to the sets {<\/span>x<\/span>\u2208<\/span>\u03a3<\/span>m<\/span><\/span>:<\/span>\u2200<\/span><\/span>k<\/span>,<\/span>\u00a0<\/span>x<\/span>k<\/span><\/span>x<\/span>2<\/span>k<\/span><\/span><\/span><\/span>.<\/span>.<\/span>.<\/span>x<\/span>n<\/span>k<\/span><\/span><\/span><\/span>=<\/span>0<\/span>}<\/span><\/span><\/span><\/span>, where n<\/span>\u2265<\/span>3<\/span><\/span><\/span><\/span>. We prove that for such sets, the Hausdorff and Minkowski dimensions typically differ.<\/p>\n","protected":false},"excerpt":{"rendered":"

We compute the Hausdorff and Minkowski dimension of subsets of the symbolic space $\\Sigma_m=\\{0,…,m-1\\}^\\N$ that are invariant under multiplication by integers. The results apply to the sets {x\u2208\u03a3m:\u2200k,\u00a0xkx2k…xnk=0}, where n\u22653. We prove that for such sets, the Hausdorff and Minkowski dimensions typically differ.<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"Cornell University Library","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"","msr_number":"","msr_organization":"","msr_pages_string":"","msr_page_range_start":"","msr_page_range_end":"","msr_series":"","msr_volume":"","msr_copyright":"","msr_conference_name":"","msr_doi":"","msr_arxiv_id":"","msr_s2_paper_id":"","msr_mag_id":"","msr_pubmed_id":"","msr_other_authors":"","msr_other_contributors":"","msr_speaker":"","msr_award":"","msr_affiliation":"","msr_institution":"","msr_host":"","msr_version":"","msr_duration":"","msr_original_fields_of_study":"","msr_release_tracker_id":"","msr_s2_match_type":"","msr_citation_count_updated":"","msr_published_date":"2012-06-21","msr_highlight_text":"","msr_notes":"","msr_longbiography":"","msr_publicationurl":"https:\/\/arxiv.org\/abs\/1206.4742","msr_external_url":"","msr_secondary_video_url":"","msr_conference_url":"","msr_journal_url":"","msr_s2_pdf_url":"","msr_year":0,"msr_citation_count":0,"msr_influential_citations":0,"msr_reference_count":0,"msr_s2_match_confidence":0,"msr_microsoftintellectualproperty":true,"msr_s2_open_access":false,"msr_s2_author_ids":[],"msr_pub_ids":[],"msr_hide_image_in_river":0,"footnotes":""},"msr-research-highlight":[],"research-area":[13546],"msr-publication-type":[193715],"msr-publisher":[],"msr-focus-area":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-357527","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-computational-sciences-mathematics","msr-locale-en_us"],"msr_publishername":"Cornell University Library","msr_edition":"","msr_affiliation":"","msr_published_date":"2012-06-21","msr_host":"","msr_duration":"","msr_version":"","msr_speaker":"","msr_other_contributors":"","msr_booktitle":"","msr_pages_string":"","msr_chapter":"","msr_isbn":"","msr_journal":"","msr_volume":"","msr_number":"","msr_editors":"","msr_series":"","msr_issue":"","msr_organization":"","msr_how_published":"","msr_notes":"","msr_highlight_text":"","msr_release_tracker_id":"","msr_original_fields_of_study":"","msr_download_urls":"","msr_external_url":"","msr_secondary_video_url":"","msr_longbiography":"","msr_microsoftintellectualproperty":1,"msr_main_download":"357530","msr_publicationurl":"https:\/\/arxiv.org\/abs\/1206.4742","msr_doi":"","msr_publication_uploader":[{"type":"file","title":"1206.4742v1","viewUrl":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-content\/uploads\/2017\/01\/1206.4742v1.pdf","id":357530,"label_id":0},{"type":"url","title":"https:\/\/arxiv.org\/abs\/1206.4742","viewUrl":false,"id":false,"label_id":0}],"msr_related_uploader":"","msr_citation_count":0,"msr_citation_count_updated":"","msr_s2_paper_id":"","msr_influential_citations":0,"msr_reference_count":0,"msr_arxiv_id":"","msr_s2_author_ids":[],"msr_s2_open_access":false,"msr_s2_pdf_url":null,"msr_attachments":[{"id":0,"url":"https:\/\/arxiv.org\/abs\/1206.4742"}],"msr-author-ordering":[{"type":"user_nicename","value":"peres","user_id":33234,"rest_url":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=peres"},{"type":"text","value":"Joerg Schmeling","user_id":0,"rest_url":false},{"type":"text","value":"St\u00e9phane Seuret","user_id":0,"rest_url":false},{"type":"text","value":"Boris Solomyak","user_id":0,"rest_url":false}],"msr_impact_theme":[],"msr_research_lab":[],"msr_event":[],"msr_group":[],"msr_project":[],"publication":[],"video":[],"msr-tool":[],"msr_publication_type":"article","related_content":[],"_links":{"self":[{"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/357527","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item"}],"about":[{"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/types\/msr-research-item"}],"version-history":[{"count":2,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/357527\/revisions"}],"predecessor-version":[{"id":416669,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/357527\/revisions\/416669"}],"wp:attachment":[{"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/media?parent=357527"}],"wp:term":[{"taxonomy":"msr-research-highlight","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-research-highlight?post=357527"},{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=357527"},{"taxonomy":"msr-publication-type","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-publication-type?post=357527"},{"taxonomy":"msr-publisher","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-publisher?post=357527"},{"taxonomy":"msr-focus-area","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-focus-area?post=357527"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=357527"},{"taxonomy":"msr-post-option","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-post-option?post=357527"},{"taxonomy":"msr-field-of-study","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-field-of-study?post=357527"},{"taxonomy":"msr-conference","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-conference?post=357527"},{"taxonomy":"msr-journal","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-journal?post=357527"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=357527"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=357527"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}