{"id":"https://openalex.org/W7224101077","doi":"https://doi.org/10.48550/arxiv.2610.35889","title":"Observable diameter of product measures with a power-type interior zero","display_name":"Observable diameter of product measures with a power-type interior zero","publication_year":2026,"publication_date":"2026-09-10","ids":{"openalex":"https://openalex.org/W7224101077","doi":"https://doi.org/10.48550/arxiv.2610.35889"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.35889","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.35889","pdf_url":"https://arxiv.org/pdf/2610.35889","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/2610.35889","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5028497380","display_name":"Takashi Shioya","orcid":"https://orcid.org/0000-0002-8981-1031","observed_orcids":["https://orcid.org/0000-0002-8981-1031"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Shioya, Takashi","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5028497380"],"corresponding_institution_ids":[],"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/T11022","display_name":"Spectral Theory in Mathematical Physics","score":0.2101999968290329,"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"}},"topics":[{"id":"https://openalex.org/T11022","display_name":"Spectral Theory in Mathematical Physics","score":0.2101999968290329,"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/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.17020000517368317,"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/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.13130000233650208,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/product-measures","display_name":"product measures","score":0.9340000152587891},{"id":"https://openalex.org/keywords/order-of-vanishing","display_name":"order of vanishing","score":0.6259999871253967},{"id":"https://openalex.org/keywords/spectral-gap","display_name":"spectral gap","score":0.8460000157356262}],"concepts":[{"id":"https://openalex.org/C32848918","wikidata":"https://www.wikidata.org/wiki/Q845789","display_name":"Observable","level":2,"score":0.7949000000953674},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7161999940872192},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.6741999983787537},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.5252000093460083},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.5169000029563904},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.5052000284194946},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.47690001130104065},{"id":"https://openalex.org/C21031990","wikidata":"https://www.wikidata.org/wiki/Q355020","display_name":"Probability measure","level":2,"score":0.42820000648498535},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.38190001249313354},{"id":"https://openalex.org/C11683690","wikidata":"https://www.wikidata.org/wiki/Q752726","display_name":"Asymptotic expansion","level":2,"score":0.3806999921798706},{"id":"https://openalex.org/C2777105136","wikidata":"https://www.wikidata.org/wiki/Q827230","display_name":"Lebesgue measure","level":3,"score":0.3481999933719635},{"id":"https://openalex.org/C197055811","wikidata":"https://www.wikidata.org/wiki/Q207522","display_name":"Probability density function","level":2,"score":0.3370000123977661},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.32580000162124634},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.314300000667572},{"id":"https://openalex.org/C205147927","wikidata":"https://www.wikidata.org/wiki/Q752718","display_name":"Asymptotic analysis","level":2,"score":0.2948000133037567},{"id":"https://openalex.org/C2987642246","wikidata":"https://www.wikidata.org/wiki/Q5452196","display_name":"First order","level":2,"score":0.2833999991416931},{"id":"https://openalex.org/C55606962","wikidata":"https://www.wikidata.org/wiki/Q1572094","display_name":"Product measure","level":3,"score":0.2648000121116638},{"id":"https://openalex.org/C168110828","wikidata":"https://www.wikidata.org/wiki/Q1331626","display_name":"Spectral density","level":2,"score":0.26350000500679016},{"id":"https://openalex.org/C156778621","wikidata":"https://www.wikidata.org/wiki/Q1365748","display_name":"Spectrum (functional analysis)","level":2,"score":0.25949999690055847},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.25839999318122864},{"id":"https://openalex.org/C123222754","wikidata":"https://www.wikidata.org/wiki/Q1070472","display_name":"Product topology","level":2,"score":0.2533999979496002},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.2524999976158142}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.35889","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.35889","pdf_url":"https://arxiv.org/pdf/2610.35889","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.2610.35889","doi":"https://doi.org/10.48550/arxiv.2610.35889","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.35889","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:2610.35889","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.35889","pdf_url":"https://arxiv.org/pdf/2610.35889","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":[{"id":"https://openalex.org/G7308168456","display_name":"\u8ddd\u96e2\u7a7a\u9593\u3068\u6e2c\u5ea6\u8ddd\u96e2\u7a7a\u9593\u306e\u5e7e\u4f55\u5b66","funder_award_id":"24K06729","funder_id":"https://openalex.org/F4320334764","funder_display_name":"Japan Society for the Promotion of Science"}],"funders":[{"id":"https://openalex.org/F4320334764","display_name":"Japan Society for the Promotion of Science","ror":"https://ror.org/00hhkn466"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7224101077.pdf","grobid_xml":"https://content.openalex.org/works/W7224101077.grobid-xml"},"referenced_works_count":2,"referenced_works":["https://openalex.org/W2065540273","https://openalex.org/W7168951561"],"related_works":["https://openalex.org/W2003576888","https://openalex.org/W4400141697","https://openalex.org/W4231539633","https://openalex.org/W890227938","https://openalex.org/W2001399686","https://openalex.org/W7220955602","https://openalex.org/W4398263950","https://openalex.org/W898219886","https://openalex.org/W2080479164","https://openalex.org/W2134178586"],"abstract_inverted_index":{"We":[0],"determine":[1],"the":[2,6,32,36,53,59,62,67,70,75,78],"asymptotic":[3,54],"order":[4,55],"of":[5,9,35,61],"observable":[7],"diameter":[8],"$\\ell^p$-product":[10],"spaces":[11],"associated":[12],"with":[13],"a":[14,22],"probability":[15],"measure":[16],"on":[17,31],"$[0,1]$":[18],"whose":[19],"density":[20],"has":[21],"power-type":[23],"interior":[24],"zero.":[25,37],"Different":[26],"vanishing":[27,64],"orders":[28],"are":[29],"allowed":[30],"two":[33,63],"sides":[34],"For":[38],"every":[39],"$1":[40],"\\le":[41,43],"p":[42],"\\infty$,":[44],"we":[45],"obtain":[46],"sharp":[47],"two-sided":[48],"estimates":[49],"and":[50],"show":[51],"that":[52],"is":[56],"governed":[57],"by":[58],"larger":[60],"orders.":[65],"In":[66],"case":[68],"$p=2$,":[69],"transition":[71],"occurs":[72],"exactly":[73],"at":[74],"threshold":[76],"for":[77],"spectral":[79],"gap.":[80]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-29T00:00:00"}
