{"id":"https://openalex.org/W7171494515","doi":"https://doi.org/10.48550/arxiv.2607.24146","title":"Nerve-type and invariance theorems for asymptotic dimension","display_name":"Nerve-type and invariance theorems for asymptotic dimension","publication_year":2026,"publication_date":"2026-07-27","ids":{"openalex":"https://openalex.org/W7171494515","doi":"https://doi.org/10.48550/arxiv.2607.24146"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2607.24146","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2607.24146","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,"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"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2607.24146","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5084900180","display_name":"Chun\u2010Hung Liu","orcid":"https://orcid.org/0000-0001-8822-9966"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Liu, Chun-Hung","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5031639997","display_name":"Sergey Norin","orcid":"https://orcid.org/0000-0003-4833-7983"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Norin, Sergey","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"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/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.32850000262260437,"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/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.32850000262260437,"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/T12536","display_name":"Topological and Geometric Data Analysis","score":0.29280000925064087,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10899","display_name":"Advanced Operator Algebra Research","score":0.07769999653100967,"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/inductive-dimension","display_name":"Inductive dimension","score":0.7964000105857849},{"id":"https://openalex.org/keywords/packing-dimension","display_name":"Packing dimension","score":0.6373999714851379},{"id":"https://openalex.org/keywords/disjoint-sets","display_name":"Disjoint sets","score":0.6244000196456909},{"id":"https://openalex.org/keywords/dimension-function","display_name":"Dimension function","score":0.6136000156402588},{"id":"https://openalex.org/keywords/intersection-graph","display_name":"Intersection graph","score":0.4569999873638153},{"id":"https://openalex.org/keywords/effective-dimension","display_name":"Effective dimension","score":0.4489000141620636},{"id":"https://openalex.org/keywords/dimension-theory","display_name":"Dimension theory (algebra)","score":0.42410001158714294},{"id":"https://openalex.org/keywords/minkowski\u2013bouligand-dimension","display_name":"Minkowski\u2013Bouligand dimension","score":0.40540000796318054},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.37869998812675476}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8439000248908997},{"id":"https://openalex.org/C66844729","wikidata":"https://www.wikidata.org/wiki/Q841816","display_name":"Inductive dimension","level":5,"score":0.7964000105857849},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6628999710083008},{"id":"https://openalex.org/C80661125","wikidata":"https://www.wikidata.org/wiki/Q5404299","display_name":"Packing dimension","level":5,"score":0.6373999714851379},{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.6244000196456909},{"id":"https://openalex.org/C28235433","wikidata":"https://www.wikidata.org/wiki/Q5277259","display_name":"Dimension function","level":3,"score":0.6136000156402588},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4634000062942505},{"id":"https://openalex.org/C54540088","wikidata":"https://www.wikidata.org/wiki/Q3498041","display_name":"Intersection graph","level":4,"score":0.4569999873638153},{"id":"https://openalex.org/C115311070","wikidata":"https://www.wikidata.org/wiki/Q5347255","display_name":"Effective dimension","level":3,"score":0.4489000141620636},{"id":"https://openalex.org/C95341145","wikidata":"https://www.wikidata.org/wiki/Q5277265","display_name":"Dimension theory (algebra)","level":3,"score":0.42410001158714294},{"id":"https://openalex.org/C101597101","wikidata":"https://www.wikidata.org/wiki/Q2555655","display_name":"Minkowski\u2013Bouligand dimension","level":4,"score":0.40540000796318054},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.37869998812675476},{"id":"https://openalex.org/C168185519","wikidata":"https://www.wikidata.org/wiki/Q5156577","display_name":"Complex dimension","level":2,"score":0.365200012922287},{"id":"https://openalex.org/C122041747","wikidata":"https://www.wikidata.org/wiki/Q838611","display_name":"Ball (mathematics)","level":2,"score":0.3434000015258789},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.33739998936653137},{"id":"https://openalex.org/C37253","wikidata":"https://www.wikidata.org/wiki/Q5156502","display_name":"Complete intersection","level":2,"score":0.3280999958515167},{"id":"https://openalex.org/C194198291","wikidata":"https://www.wikidata.org/wiki/Q565186","display_name":"Hausdorff dimension","level":2,"score":0.320499986410141},{"id":"https://openalex.org/C198043062","wikidata":"https://www.wikidata.org/wiki/Q180953","display_name":"Metric space","level":2,"score":0.31839999556541443},{"id":"https://openalex.org/C23707678","wikidata":"https://www.wikidata.org/wiki/Q164262","display_name":"Lebesgue covering dimension","level":3,"score":0.31630000472068787},{"id":"https://openalex.org/C60933471","wikidata":"https://www.wikidata.org/wiki/Q17152964","display_name":"Metric dimension","level":5,"score":0.31349998712539673},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.3012000024318695},{"id":"https://openalex.org/C2780012671","wikidata":"https://www.wikidata.org/wiki/Q1343870","display_name":"Corollary","level":2,"score":0.2976999878883362},{"id":"https://openalex.org/C64543145","wikidata":"https://www.wikidata.org/wiki/Q162942","display_name":"Intersection (aeronautics)","level":2,"score":0.29510000348091125},{"id":"https://openalex.org/C150591576","wikidata":"https://www.wikidata.org/wiki/Q739925","display_name":"Family of sets","level":3,"score":0.28630000352859497},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.2759000062942505},{"id":"https://openalex.org/C196613150","wikidata":"https://www.wikidata.org/wiki/Q3966092","display_name":"Convex metric space","level":3,"score":0.257999986410141},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.25519999861717224}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2607.24146","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2607.24146","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,"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":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2607.24146","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2607.24146","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,"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"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Asymptotic":[0],"dimension":[1,11,52,65,99,130,161,201,226,251],"of":[2,9,12,18,21,31,40,53,57,66,73,91,93,97,105,115,118,131,135,156,162,166,169,173,184,202,206,211,215,227,231,234,237,252,256,258],"metric":[3,69,95],"spaces":[4],"is":[5,23,88,138,144,189],"a":[6,19,58,89,94,154,182,207,216,244],"large-scale":[7],"analog":[8],"covering":[10],"topological":[13,218],"spaces.":[14],"An":[15],"intersection":[16,55,133,164,204,229,254],"graph":[17,25,56,134,165,205,230],"family":[20,33,59,90,168,183,208,233],"sets":[22,172,214,239],"the":[24,29,32,50,54,63,67,128,132,159,163,199,203,224,228,232,235,238,249,253],"whose":[26,35],"vertices":[27],"are":[28],"members":[30,41,72,114],"and":[34,62,79,148],"edges":[36],"correspond":[37],"to":[38],"pairs":[39],"with":[42,220],"non-empty":[43],"intersection.":[44],"Our":[45,193],"first":[46],"main":[47,195],"result":[48,143,196],"connects":[49],"asymptotic":[51,129,160,200,225,250],"${\\mathcal":[60,74,86,116,136,209,241],"F}$":[61,75,87,117,137,210],"Assouad-Nagata":[64,98],"ambient":[68],"space":[70,96,219],"containing":[71],"under":[76,243],"some":[77,124],"mild":[78,245],"necessary":[80],"assumptions.":[81],"We":[82],"prove":[83],"that":[84,102,198],"if":[85],"subsets":[92],"$n$":[100,264],"such":[101,180],"every":[103],"ball":[104],"radius":[106],"$r$":[107],"intersects":[108],"at":[109,120,139,190],"most":[110,140,191],"$f(r/s)$":[111],"pairwise":[112],"disjoint":[113],"diameter":[119],"least":[121],"$s$":[122],"for":[123],"function":[125],"$f$,":[126],"then":[127],"$n+1$.":[141,192],"This":[142],"optimal":[145],"both":[146],"quantitatively":[147],"qualitatively":[149],"in":[150,177,186,240,260],"several":[151],"senses.":[152],"As":[153],"corollary":[155],"this":[157],"result,":[158],"any":[167],"compact":[170],"convex":[171],"bounded":[174],"aspect":[175],"ratio":[176],"${\\mathbb":[178,187,261],"R}^n$,":[179,188],"as":[181],"balls":[185],"second":[194],"states":[197],"connected":[212,217,221],"closed":[213],"boundary":[222,236],"equals":[223,263],"F}$,":[242],"condition.":[246],"In":[247],"particular,":[248],"graphs":[255],"families":[257],"spheres":[259],"R}^n$":[262],"or":[265],"$n+1$":[266],"when":[267],"$n":[268],"\\geq":[269],"2$.":[270]},"counts_by_year":[],"updated_date":"2026-07-29T09:40:50.615796","created_date":"2026-07-29T00:00:00"}