{"id":"https://openalex.org/W7221069123","doi":"https://doi.org/10.48550/arxiv.2610.12351","title":"Prevalence of sparsity for negatively curved metrics","display_name":"Prevalence of sparsity for negatively curved metrics","publication_year":2026,"publication_date":"2026-10-08","ids":{"openalex":"https://openalex.org/W7221069123","doi":"https://doi.org/10.48550/arxiv.2610.12351"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.12351","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.12351","pdf_url":"https://arxiv.org/pdf/2610.12351","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/2610.12351","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5075441789","display_name":"Kostiantyn Drach","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I4210122226","display_name":"Centre de Recerca Matem\u00e0tica","ror":"https://ror.org/020s51w82","country_code":"ES","type":"facility","lineage":["https://openalex.org/I4210122226","https://openalex.org/I4387153040"]},{"id":"https://openalex.org/I71999127","display_name":"Universitat de Barcelona","ror":"https://ror.org/021018s57","country_code":"ES","type":"education","lineage":["https://openalex.org/I71999127"]}],"countries":["ES"],"is_corresponding":false,"raw_author_name":"Drach, Kostiantyn","raw_affiliation_strings":["Universitat de Barcelona , Gran Via de les Corts Catalanes , 585 , 08007 Barcelona , Spain Centre de Recerca Matem\u00e0tica , Edifici C , Carrer de l'Albareda , 08193 Bellaterra , Spain"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Universitat de Barcelona , Gran Via de les Corts Catalanes , 585 , 08007 Barcelona , Spain Centre de Recerca Matem\u00e0tica , Edifici C , Carrer de l'Albareda , 08193 Bellaterra , Spain","institution_ids":["https://openalex.org/I4210122226","https://openalex.org/I71999127"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5158187506","display_name":"Vadim Kaloshin","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I157556583","display_name":"Institute of Science and Technology Austria","ror":"https://ror.org/03gnh5541","country_code":"AT","type":"education","lineage":["https://openalex.org/I157556583"]},{"id":"https://openalex.org/I4210122226","display_name":"Centre de Recerca Matem\u00e0tica","ror":"https://ror.org/020s51w82","country_code":"ES","type":"facility","lineage":["https://openalex.org/I4210122226","https://openalex.org/I4387153040"]},{"id":"https://openalex.org/I71999127","display_name":"Universitat de Barcelona","ror":"https://ror.org/021018s57","country_code":"ES","type":"education","lineage":["https://openalex.org/I71999127"]}],"countries":["AT","ES"],"is_corresponding":false,"raw_author_name":"Kaloshin, Vadim","raw_affiliation_strings":["Institute of Science and Technology Austria , Am Campus 1 , 3400 Klosterneuburg , Austria","Universitat de Barcelona , Gran Via de les Corts Catalanes , 585 , 08007 Barcelona , Spain Centre de Recerca Matem\u00e0tica , Edifici C , Carrer de l'Albareda , 08193 Bellaterra , Spain"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Science and Technology Austria , Am Campus 1 , 3400 Klosterneuburg , Austria","institution_ids":["https://openalex.org/I157556583"]},{"raw_affiliation_string":"Universitat de Barcelona , Gran Via de les Corts Catalanes , 585 , 08007 Barcelona , Spain Centre de Recerca Matem\u00e0tica , Edifici C , Carrer de l'Albareda , 08193 Bellaterra , Spain","institution_ids":["https://openalex.org/I4210122226","https://openalex.org/I71999127"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":[],"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/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.7124000191688538,"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":0.7124000191688538,"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":0.10109999775886536,"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"}},{"id":"https://openalex.org/T13080","display_name":"Advanced Differential Geometry Research","score":0.06270000338554382,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/negatively-curved-metrics","display_name":"negatively curved metrics","score":0.7990000247955322},{"id":"https://openalex.org/keywords/length-spectrum","display_name":"length spectrum","score":0.906000018119812},{"id":"https://openalex.org/keywords/closed-geodesics","display_name":"closed geodesics","score":0.8069999814033508},{"id":"https://openalex.org/keywords/prevalence","display_name":"prevalence","score":0.6499999761581421},{"id":"https://openalex.org/keywords/expanding-circle-maps","display_name":"expanding circle maps","score":0.7519999742507935},{"id":"https://openalex.org/keywords/local-rigidity","display_name":"local rigidity","score":0.8529999852180481},{"id":"https://openalex.org/keywords/negatively-curved-manifolds","display_name":"negatively curved manifolds","score":0.8140000104904175}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7699000239372253},{"id":"https://openalex.org/C165818556","wikidata":"https://www.wikidata.org/wiki/Q213488","display_name":"Geodesic","level":2,"score":0.734000027179718},{"id":"https://openalex.org/C75235859","wikidata":"https://www.wikidata.org/wiki/Q582659","display_name":"Exponential growth","level":2,"score":0.6438999772071838},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.6129000186920166},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.5472999811172485},{"id":"https://openalex.org/C156778621","wikidata":"https://www.wikidata.org/wiki/Q1365748","display_name":"Spectrum (functional analysis)","level":2,"score":0.5184999704360962},{"id":"https://openalex.org/C2780388253","wikidata":"https://www.wikidata.org/wiki/Q5421508","display_name":"Exponent","level":2,"score":0.4950999915599823},{"id":"https://openalex.org/C529865628","wikidata":"https://www.wikidata.org/wiki/Q1790740","display_name":"Manifold (fluid mechanics)","level":2,"score":0.4754999876022339},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.4097999930381775},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.4081000089645386},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.39469999074935913},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.392300009727478},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.38199999928474426},{"id":"https://openalex.org/C160343418","wikidata":"https://www.wikidata.org/wiki/Q185256","display_name":"Rigidity (electromagnetism)","level":2,"score":0.31150001287460327},{"id":"https://openalex.org/C198043062","wikidata":"https://www.wikidata.org/wiki/Q180953","display_name":"Metric space","level":2,"score":0.29429998993873596},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.29269999265670776},{"id":"https://openalex.org/C22251595","wikidata":"https://www.wikidata.org/wiki/Q670036","display_name":"Arc length","level":3,"score":0.266400009393692},{"id":"https://openalex.org/C163415756","wikidata":"https://www.wikidata.org/wiki/Q126473","display_name":"Contraction (grammar)","level":2,"score":0.2660999894142151}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.12351","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.12351","pdf_url":"https://arxiv.org/pdf/2610.12351","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.2610.12351","doi":"https://doi.org/10.48550/arxiv.2610.12351","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.12351","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.12351","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.12351","pdf_url":"https://arxiv.org/pdf/2610.12351","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/G1317067511","display_name":null,"funder_award_id":"PID2023-147252NB-I00","funder_id":"https://openalex.org/F4320335598","funder_display_name":"Agencia Estatal de Investigaci\u00f3n"},{"id":"https://openalex.org/G4729193157","display_name":null,"funder_award_id":"CNS2025-166633","funder_id":"https://openalex.org/F4320335598","funder_display_name":"Agencia Estatal de Investigaci\u00f3n"},{"id":"https://openalex.org/G6325762137","display_name":null,"funder_award_id":"CEX2020-001084-M","funder_id":"https://openalex.org/F4320321505","funder_display_name":"Generalitat de Catalunya"},{"id":"https://openalex.org/G7013227978","display_name":null,"funder_award_id":"2021 SGR 00697","funder_id":"https://openalex.org/F4320321505","funder_display_name":"Generalitat de Catalunya"},{"id":"https://openalex.org/G84477964","display_name":null,"funder_award_id":"CEX2020-001084-M","funder_id":"https://openalex.org/F4320335598","funder_display_name":"Agencia Estatal de Investigaci\u00f3n"}],"funders":[{"id":"https://openalex.org/F4320321505","display_name":"Generalitat de Catalunya","ror":"https://ror.org/01bg62x04"},{"id":"https://openalex.org/F4320335598","display_name":"Agencia Estatal de Investigaci\u00f3n","ror":"https://ror.org/003x0zc53"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7221069123.pdf","grobid_xml":"https://content.openalex.org/works/W7221069123.grobid-xml"},"referenced_works_count":7,"referenced_works":["https://openalex.org/W2100299994","https://openalex.org/W2282118389","https://openalex.org/W2808514814","https://openalex.org/W2999318180","https://openalex.org/W4416104262","https://openalex.org/W4416599775","https://openalex.org/W7214129301"],"related_works":["https://openalex.org/W2808514814","https://openalex.org/W7214129301","https://openalex.org/W2952338448","https://openalex.org/W4378345359","https://openalex.org/W4387963950","https://openalex.org/W2228301608","https://openalex.org/W4390813719","https://openalex.org/W2187825058","https://openalex.org/W4414744721","https://openalex.org/W1642011263"],"abstract_inverted_index":{"Let":[0],"$M$":[1,19,70],"be":[2],"a":[3,13,52,112],"closed":[4,28],"manifold":[5],"of":[6,12,23,25,30],"arbitrary":[7],"dimension.":[8],"The":[9],"length":[10,37,104],"spectrum":[11,38],"negatively":[14,66,124],"curved":[15,67,125],"metric":[16],"$g$":[17],"on":[18,69,101],"is":[20,39,95],"the":[21,43,57,81,90],"set":[22],"lengths":[24,47],"all":[26,78],"possible":[27],"geodesics":[29],"$g$.":[31],"We":[32],"prove":[33],"that":[34,122],"metrics":[35,68,126],"whose":[36],"exponentially":[40,55,128],"sparse,":[41],"i.e.,":[42],"gaps":[44],"between":[45],"distinct":[46],"are":[48,59,131],"bounded":[49],"below":[50],"by":[51,97,116],"quantity":[53],"decaying":[54],"in":[56,77,83,89],"length,":[58],"prevalent,":[60],"and":[61,111,120],"hence":[62],"dense,":[63],"among":[64],"$C^k$-smooth":[65,123],"for":[71,107],"sufficiently":[72],"large":[73],"$k$.":[74,92],"Moreover,":[75],"unlike":[76],"known":[79],"constructions,":[80],"exponent":[82],"our":[84,98],"bound":[85],"grows":[86],"only":[87],"sublinearly":[88],"regularity":[91],"This":[93],"work":[94],"motivated":[96],"recent":[99,114],"result":[100,115],"prevalent":[102],"unmarked":[103],"spectral":[105],"rigidity":[106],"expanding":[108],"circle":[109],"maps":[110],"very":[113],"DeWitt,":[117],"Durham,":[118],"Reber,":[119],"O'Hare":[121],"with":[127],"sparse":[129],"spectra":[130],"locally":[132],"rigid.":[133]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-10T00:00:00"}
