{"id":"https://openalex.org/W7220953777","doi":"https://doi.org/10.48550/arxiv.2610.08549","title":"The additivity of a certain Hausdorff measure can differ from that of the Lebesgue measure","display_name":"The additivity of a certain Hausdorff measure can differ from that of the Lebesgue measure","publication_year":2026,"publication_date":"2026-10-06","ids":{"openalex":"https://openalex.org/W7220953777","doi":"https://doi.org/10.48550/arxiv.2610.08549"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.08549","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.08549","pdf_url":"https://arxiv.org/pdf/2610.08549","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2610.08549","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5004549780","display_name":"Tatsuya Goto","orcid":"https://orcid.org/0009-0006-4732-9614","observed_orcids":["https://orcid.org/0009-0006-4732-9614"]},"institutions":[{"id":"https://openalex.org/I145847075","display_name":"TU Wien","ror":"https://ror.org/04d836q62","country_code":"AT","type":"education","lineage":["https://openalex.org/I145847075"]}],"countries":["AT"],"is_corresponding":true,"raw_author_name":"Goto, Tatsuya","raw_affiliation_strings":["Institute of Discrete Mathematics and Geometry , TU Wien Wiedner Hauptstrasse 8-10/104 , 1040 Wien , Austria"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Discrete Mathematics and Geometry , TU Wien Wiedner Hauptstrasse 8-10/104 , 1040 Wien , Austria","institution_ids":["https://openalex.org/I145847075"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5004549780"],"corresponding_institution_ids":["https://openalex.org/I145847075"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11151","display_name":"Advanced Topology and Set Theory","score":0.98580002784729,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11151","display_name":"Advanced Topology and Set Theory","score":0.98580002784729,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.0024999999441206455,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T13400","display_name":"Mathematical and Theoretical Analysis","score":0.0015999999595806003,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/hausdorff-measure","display_name":"Hausdorff measure","score":0.9819999933242798},{"id":"https://openalex.org/keywords/null-ideal","display_name":"null ideal","score":0.6959999799728394},{"id":"https://openalex.org/keywords/additivity","display_name":"additivity","score":0.6100000143051147},{"id":"https://openalex.org/keywords/cofinality","display_name":"cofinality","score":0.9959999918937683},{"id":"https://openalex.org/keywords/consistency-results","display_name":"consistency results","score":0.6050000190734863},{"id":"https://openalex.org/keywords/localization-system","display_name":"localization system","score":0.8949999809265137},{"id":"https://openalex.org/keywords/finite-graphs","display_name":"finite graphs","score":0.5400000214576721},{"id":"https://openalex.org/keywords/cardinal-characteristics","display_name":"cardinal characteristics","score":0.4189999997615814},{"id":"https://openalex.org/keywords/bounding-number","display_name":"bounding number","score":0.3319999873638153},{"id":"https://openalex.org/keywords/covering-number","display_name":"covering number","score":0.718999981880188},{"id":"https://openalex.org/keywords/consistency-strength","display_name":"consistency strength","score":0.6629999876022339}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8672999739646912},{"id":"https://openalex.org/C2777105136","wikidata":"https://www.wikidata.org/wiki/Q827230","display_name":"Lebesgue measure","level":3,"score":0.652400016784668},{"id":"https://openalex.org/C148070024","wikidata":"https://www.wikidata.org/wiki/Q1283623","display_name":"Cofinality","level":4,"score":0.598800003528595},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.5837000012397766},{"id":"https://openalex.org/C73225184","wikidata":"https://www.wikidata.org/wiki/Q1591095","display_name":"Hausdorff measure","level":3,"score":0.5460000038146973},{"id":"https://openalex.org/C102065156","wikidata":"https://www.wikidata.org/wiki/Q291430","display_name":"\u03c3-finite measure","level":4,"score":0.5453000068664551},{"id":"https://openalex.org/C14158598","wikidata":"https://www.wikidata.org/wiki/Q853831","display_name":"Lebesgue integration","level":2,"score":0.5271000266075134},{"id":"https://openalex.org/C191399826","wikidata":"https://www.wikidata.org/wiki/Q326908","display_name":"Hausdorff space","level":2,"score":0.5034000277519226},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.49300000071525574},{"id":"https://openalex.org/C38803922","wikidata":"https://www.wikidata.org/wiki/Q95745348","display_name":"Additive function","level":2,"score":0.4902999997138977},{"id":"https://openalex.org/C141898687","wikidata":"https://www.wikidata.org/wiki/Q1501997","display_name":"Hausdorff distance","level":2,"score":0.42989999055862427},{"id":"https://openalex.org/C2776639384","wikidata":"https://www.wikidata.org/wiki/Q840396","display_name":"Ideal (ethics)","level":2,"score":0.4171000123023987},{"id":"https://openalex.org/C203763787","wikidata":"https://www.wikidata.org/wiki/Q371029","display_name":"Null (SQL)","level":2,"score":0.4023999869823456},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3939000070095062},{"id":"https://openalex.org/C43994743","wikidata":"https://www.wikidata.org/wiki/Q516776","display_name":"Measurable function","level":3,"score":0.3779999911785126},{"id":"https://openalex.org/C55375708","wikidata":"https://www.wikidata.org/wiki/Q1201815","display_name":"Null set","level":3,"score":0.37040001153945923},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3594000041484833},{"id":"https://openalex.org/C2776436953","wikidata":"https://www.wikidata.org/wiki/Q5163215","display_name":"Consistency (knowledge bases)","level":2,"score":0.34779998660087585},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.3359000086784363},{"id":"https://openalex.org/C143924541","wikidata":"https://www.wikidata.org/wiki/Q258374","display_name":"Outer measure","level":5,"score":0.33410000801086426},{"id":"https://openalex.org/C166758865","wikidata":"https://www.wikidata.org/wiki/Q892585","display_name":"Borel measure","level":3,"score":0.33070001006126404},{"id":"https://openalex.org/C189950617","wikidata":"https://www.wikidata.org/wiki/Q937228","display_name":"Property (philosophy)","level":2,"score":0.3278000056743622},{"id":"https://openalex.org/C194198291","wikidata":"https://www.wikidata.org/wiki/Q565186","display_name":"Hausdorff dimension","level":2,"score":0.27410000562667847},{"id":"https://openalex.org/C162838799","wikidata":"https://www.wikidata.org/wiki/Q596077","display_name":"Counterexample","level":2,"score":0.2687999904155731},{"id":"https://openalex.org/C111335779","wikidata":"https://www.wikidata.org/wiki/Q3454686","display_name":"Reduction (mathematics)","level":2,"score":0.26409998536109924},{"id":"https://openalex.org/C118733216","wikidata":"https://www.wikidata.org/wiki/Q332504","display_name":"Absolute continuity","level":2,"score":0.25450000166893005},{"id":"https://openalex.org/C17137986","wikidata":"https://www.wikidata.org/wiki/Q215067","display_name":"Orthogonality","level":2,"score":0.25060001015663147}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.08549","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.08549","pdf_url":"https://arxiv.org/pdf/2610.08549","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2610.08549","doi":"https://doi.org/10.48550/arxiv.2610.08549","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.08549","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.08549","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.08549","pdf_url":"https://arxiv.org/pdf/2610.08549","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7220953777.pdf","grobid_xml":"https://content.openalex.org/works/W7220953777.grobid-xml"},"referenced_works_count":1,"referenced_works":["https://openalex.org/W7126073558"],"related_works":["https://openalex.org/W2136638155","https://openalex.org/W7126073558","https://openalex.org/W2205267289","https://openalex.org/W2148932839","https://openalex.org/W7098292068","https://openalex.org/W7099360476","https://openalex.org/W4408529950","https://openalex.org/W4232282788","https://openalex.org/W2095210755","https://openalex.org/W2139341947"],"abstract_inverted_index":{"Let":[0],"$\\mathcal{N}^h_\u03a9$":[1,58],"be":[2,80],"the":[3,7,45,53,64,72,76],"null":[4,66],"ideal":[5],"of":[6,20,47,57,63],"Hausdorff":[8],"measure":[9],"constructed":[10],"by":[11],"Davies":[12],"and":[13,34,55],"Rogers.":[14],"We":[15,68],"give":[16],"a":[17,23],"Tukey":[18],"reduction":[19],"$(\\mathcal{N}^h_\u03a9,\\mathcal{N}^h_\u03a9,\\subseteq)$":[21],"to":[22],"localization":[24],"system":[25],"with":[26],"finite":[27,73],"coordinate":[28],"sets.":[29],"It":[30],"follows":[31],"that":[32,71,83],"$\\mathfrak{v}^\\forall_{D,g}\\le\\operatorname{add}(\\mathcal{N}^h_\u03a9)$":[33],"$\\operatorname{cof}(\\mathcal{N}^h_\u03a9)\\le\\mathfrak{c}^\\forall_{D,g}$":[35],"for":[36],"some":[37],"$D,":[38],"g":[39],"\\in":[40],"\u03c9^\u03c9$.":[41],"Consequently,":[42],"we":[43],"prove":[44],"consistency":[46],"$\\mathfrak{d}<\\operatorname{add}(\\mathcal{N}^h_\u03a9)$":[48],"and,":[49],"separately,":[50],"$\\operatorname{cof}(\\mathcal{N}^h_\u03a9)<\\mathfrak{b}$.":[51],"Hence":[52],"additivity":[54],"cofinality":[56],"can":[59,79],"differ":[60],"from":[61],"those":[62],"Lebesgue":[65],"ideal.":[67],"also":[69],"show":[70],"graphs":[74],"in":[75],"Davies--Rogers":[77],"construction":[78],"chosen":[81],"so":[82],"$\\operatorname{cov}(\\mathcal{N}^h_\u03a9)\\le\\operatorname{non}(\\mathcal{N}^h_\u03a9)$.":[84]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-09T00:00:00"}
