{"id":"https://openalex.org/W7214198570","doi":"https://doi.org/10.48550/arxiv.2609.26950","title":"Geometric mean quantization via adaptive approximation","display_name":"Geometric mean quantization via adaptive approximation","publication_year":2026,"publication_date":"2026-09-22","ids":{"openalex":"https://openalex.org/W7214198570","doi":"https://doi.org/10.48550/arxiv.2609.26950"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.26950","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.26950","pdf_url":"https://arxiv.org/pdf/2609.26950","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/2609.26950","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5028805962","display_name":"Marc Ke\u00dfeb\u00f6hmer","orcid":"https://orcid.org/0000-0003-3156-4522","observed_orcids":["https://orcid.org/0000-0003-3156-4522"]},"institutions":[{"id":"https://openalex.org/I180437899","display_name":"University of Bremen","ror":"https://ror.org/04ers2y35","country_code":"DE","type":"education","lineage":["https://openalex.org/I180437899"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Kesseb\u00f6hmer, Marc","raw_affiliation_strings":["Institute for Dynamical Systems , FB -Mathematics and Computer Science , University of Bremen , Bibliothekstr. 5 , 28359 Bremen , Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute for Dynamical Systems , FB -Mathematics and Computer Science , University of Bremen , Bibliothekstr. 5 , 28359 Bremen , Germany","institution_ids":["https://openalex.org/I180437899"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5041449920","display_name":"Aljoscha Niemann","orcid":"https://orcid.org/0000-0003-4484-7919","observed_orcids":["https://orcid.org/0000-0003-4484-7919"]},"institutions":[{"id":"https://openalex.org/I180437899","display_name":"University of Bremen","ror":"https://ror.org/04ers2y35","country_code":"DE","type":"education","lineage":["https://openalex.org/I180437899"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Niemann, Aljoscha","raw_affiliation_strings":["Institute for Dynamical Systems , FB -Mathematics and Computer Science , University of Bremen , Bibliothekstr. 5 , 28359 Bremen , Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute for Dynamical Systems , FB -Mathematics and Computer Science , University of Bremen , Bibliothekstr. 5 , 28359 Bremen , Germany","institution_ids":["https://openalex.org/I180437899"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I180437899"],"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":0.8374000191688538,"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":0.8374000191688538,"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/T13234","display_name":"p-Adic Analysis and Mathematical Physics","score":0.08560000360012054,"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":0.024000000208616257,"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/quantization-dimension","display_name":"quantization dimension","score":0.6110000014305115},{"id":"https://openalex.org/keywords/local-dimension","display_name":"local dimension","score":0.4309999942779541},{"id":"https://openalex.org/keywords/entropy-dimension","display_name":"entropy dimension","score":0.6970000267028809},{"id":"https://openalex.org/keywords/weak-convergence","display_name":"weak convergence","score":0.8560000061988831},{"id":"https://openalex.org/keywords/regime-switching-models","display_name":"regime switching","score":0.6830000281333923},{"id":"https://openalex.org/keywords/self-similarity","display_name":"self-similarity","score":0.5590000152587891},{"id":"https://openalex.org/keywords/fractal-geometry","display_name":"fractal geometry","score":0.4169999957084656}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8948000073432922},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.630299985408783},{"id":"https://openalex.org/C203858083","wikidata":"https://www.wikidata.org/wiki/Q188347","display_name":"Harmonic mean","level":2,"score":0.5641000270843506},{"id":"https://openalex.org/C106301342","wikidata":"https://www.wikidata.org/wiki/Q4117933","display_name":"Entropy (arrow of time)","level":2,"score":0.48399999737739563},{"id":"https://openalex.org/C152361515","wikidata":"https://www.wikidata.org/wiki/Q181328","display_name":"Bernoulli's principle","level":2,"score":0.47749999165534973},{"id":"https://openalex.org/C28855332","wikidata":"https://www.wikidata.org/wiki/Q198099","display_name":"Quantization (signal processing)","level":2,"score":0.4681999981403351},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.4577000141143799},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4523000121116638},{"id":"https://openalex.org/C2778114796","wikidata":"https://www.wikidata.org/wiki/Q7575194","display_name":"Spectral gap","level":2,"score":0.44760000705718994},{"id":"https://openalex.org/C185767445","wikidata":"https://www.wikidata.org/wiki/Q207952","display_name":"Law of large numbers","level":3,"score":0.41850000619888306},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.3779999911785126},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.3540000021457672},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.35089999437332153},{"id":"https://openalex.org/C60432849","wikidata":"https://www.wikidata.org/wiki/Q2072112","display_name":"Unit cube","level":2,"score":0.3492000102996826},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.3458000123500824},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.3278999924659729},{"id":"https://openalex.org/C21031990","wikidata":"https://www.wikidata.org/wiki/Q355020","display_name":"Probability measure","level":2,"score":0.3249000012874603},{"id":"https://openalex.org/C95763700","wikidata":"https://www.wikidata.org/wiki/Q578985","display_name":"Convergence of random variables","level":3,"score":0.31380000710487366},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.3009999990463257},{"id":"https://openalex.org/C9679016","wikidata":"https://www.wikidata.org/wiki/Q1417473","display_name":"Principle of maximum entropy","level":2,"score":0.2881999909877777},{"id":"https://openalex.org/C53051483","wikidata":"https://www.wikidata.org/wiki/Q861555","display_name":"Cube (algebra)","level":2,"score":0.28780001401901245},{"id":"https://openalex.org/C40222840","wikidata":"https://www.wikidata.org/wiki/Q185049","display_name":"Geometric mean","level":2,"score":0.2808000147342682},{"id":"https://openalex.org/C7321624","wikidata":"https://www.wikidata.org/wiki/Q205","display_name":"Infinity","level":2,"score":0.2759000062942505},{"id":"https://openalex.org/C166785042","wikidata":"https://www.wikidata.org/wiki/Q190391","display_name":"Central limit theorem","level":2,"score":0.27489998936653137},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.26989999413490295},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.26910001039505005},{"id":"https://openalex.org/C84498859","wikidata":"https://www.wikidata.org/wiki/Q5158327","display_name":"Concentration of measure","level":2,"score":0.267300009727478},{"id":"https://openalex.org/C2778049214","wikidata":"https://www.wikidata.org/wiki/Q7512234","display_name":"Sigma","level":2,"score":0.2587999999523163},{"id":"https://openalex.org/C122203268","wikidata":"https://www.wikidata.org/wiki/Q5862903","display_name":"Probability theory","level":2,"score":0.2574999928474426},{"id":"https://openalex.org/C57945734","wikidata":"https://www.wikidata.org/wiki/Q7977941","display_name":"Weak convergence","level":3,"score":0.2524000108242035}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.26950","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.26950","pdf_url":"https://arxiv.org/pdf/2609.26950","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.2609.26950","doi":"https://doi.org/10.48550/arxiv.2609.26950","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.26950","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:2609.26950","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.26950","pdf_url":"https://arxiv.org/pdf/2609.26950","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/W7214198570.pdf","grobid_xml":"https://content.openalex.org/works/W7214198570.grobid-xml"},"referenced_works_count":10,"referenced_works":["https://openalex.org/W1966992751","https://openalex.org/W2010516941","https://openalex.org/W2072769450","https://openalex.org/W2112136864","https://openalex.org/W2221292279","https://openalex.org/W2890925093","https://openalex.org/W3043900114","https://openalex.org/W4221162822","https://openalex.org/W4281868153","https://openalex.org/W4416133386"],"related_works":["https://openalex.org/W2010516941","https://openalex.org/W1966992751","https://openalex.org/W1910768613","https://openalex.org/W4281868153","https://openalex.org/W1868022033","https://openalex.org/W2221292279","https://openalex.org/W2117588157","https://openalex.org/W6979343180","https://openalex.org/W4408346636","https://openalex.org/W2189140204"],"abstract_inverted_index":{"Let":[0],"$\u03bd$":[1,53],"be":[2,33],"a":[3,19,104,125,187],"compactly":[4],"supported":[5],"Borel":[6],"probability":[7],"measure":[8],"on":[9],"$\\mathbb":[10],"R^{d}$":[11],"with":[12,181],"$\u03bd(B(x,r))\\leq":[13],"Cr^{a}$":[14],"for":[15],"some":[16],"$a>0$.":[17],"Refine":[18],"dyadic":[20,71],"cube":[21],"exactly":[22],"when":[23,66],"its":[24],"mass":[25],"is":[26,78,116],"at":[27,37],"least":[28],"$t$,":[29],"and":[30,47,57,76,97,100,139,147,155,172,186],"let":[31],"$\\mathcal{L}_\u03bd(t)$":[32],"the":[34,45,55,94,112,121,132,137,140,144,169],"mean":[35,142],"depth":[36],"which":[38],"this":[39,82],"refinement":[40],"stops.":[41],"We":[42],"show":[43],"that":[44],"lower":[46,56,171],"upper":[48,58,173],"geometric-mean":[49],"quantization":[50,101],"dimensions":[51,96,102,134],"of":[52,60,93,111,120,127,143,168],"are":[54,135],"limits":[59],"$\\log(1/t)/\\mathcal{L}_\u03bd(t)$.":[61],"The":[62,84],"dimension":[63],"exists":[64],"precisely":[65],"$(q-1)\\sum_{Q}\u03bd(Q)^{q}$,":[67],"summed":[68],"over":[69],"all":[70],"cubes,":[72],"converges":[73],"as":[74],"$q\\downarrow1$,":[75],"it":[77],"then":[79,136],"determined":[80],"by":[81,152],"limit.":[83],"mass-threshold":[85],"formula":[86],"yields":[87],"harmonic":[88,141],"integral":[89],"bounds":[90,154],"in":[91,103,109,124,184,192],"terms":[92],"local":[95,113],"encloses":[98],"entropy":[99],"common":[105],"spectral":[106],"interval.":[107],"Convergence":[108],"law":[110,180,193],"information":[114],"rates":[115],"equivalent":[117],"to":[118],"convergence":[119,160,191],"rescaled":[122],"spectra":[123],"window":[126],"width":[128],"$1/k$":[129],"around":[130],"$q=1$;":[131],"two":[133],"arithmetic":[138],"limit":[145],"law,":[146],"we":[148],"quantify":[149],"their":[150],"difference":[151],"sharp":[153],"variance":[156,164],"identities.":[157],"Without":[158],"any":[159],"assumption,":[161],"vanishing":[162],"threshold":[163],"still":[165],"forces":[166],"equality":[167],"corresponding":[170],"dimensions.":[174],"Bernoulli":[175],"mixtures":[176],"realise":[177],"every":[178],"local-dimension":[179],"compact":[182],"support":[183],"$(0,1]$,":[185],"regime-switching":[188],"example":[189],"separates":[190],"from":[194],"almost-everywhere":[195],"convergence.":[196]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-25T00:00:00"}
