{"id":"https://openalex.org/W7220693539","doi":"https://doi.org/10.48550/arxiv.2610.04395","title":"Projection theorems for macroscopic intermediate dimensions","display_name":"Projection theorems for macroscopic intermediate dimensions","publication_year":2026,"publication_date":"2026-10-03","ids":{"openalex":"https://openalex.org/W7220693539","doi":"https://doi.org/10.48550/arxiv.2610.04395"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.04395","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.04395","pdf_url":"https://arxiv.org/pdf/2610.04395","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/2610.04395","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5140011622","display_name":"Jinjun Li","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Li, Jinjun","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5022983572","display_name":"\u595a\u674e\u5cf0","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Xi, Lifeng","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":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":0.9526000022888184,"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":0.9526000022888184,"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/T11049","display_name":"Advanced Harmonic Analysis Research","score":0.010099999606609344,"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/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.009600000455975533,"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/projection-theorem","display_name":"projection theorem","score":0.8619999885559082},{"id":"https://openalex.org/keywords/fractional-moments","display_name":"fractional moments","score":0.46000000834465027},{"id":"https://openalex.org/keywords/quasi-isometry","display_name":"quasi-isometry","score":0.7670000195503235},{"id":"https://openalex.org/keywords/hausdorff-dimension","display_name":"Hausdorff dimension","score":0.8050000071525574},{"id":"https://openalex.org/keywords/self-similar-sets","display_name":"self-similar sets","score":0.9610000252723694},{"id":"https://openalex.org/keywords/lipschitz-graphs","display_name":"Lipschitz graphs","score":0.9800000190734863},{"id":"https://openalex.org/keywords/borel-sets","display_name":"Borel sets","score":0.7360000014305115}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.807699978351593},{"id":"https://openalex.org/C194198291","wikidata":"https://www.wikidata.org/wiki/Q565186","display_name":"Hausdorff dimension","level":2,"score":0.7560999989509583},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.5979999899864197},{"id":"https://openalex.org/C12362212","wikidata":"https://www.wikidata.org/wiki/Q728435","display_name":"Linear subspace","level":2,"score":0.5906999707221985},{"id":"https://openalex.org/C80661125","wikidata":"https://www.wikidata.org/wiki/Q5404299","display_name":"Packing dimension","level":5,"score":0.5813999772071838},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.508899986743927},{"id":"https://openalex.org/C57493831","wikidata":"https://www.wikidata.org/wiki/Q3134666","display_name":"Projection (relational algebra)","level":2,"score":0.5059000253677368},{"id":"https://openalex.org/C22324862","wikidata":"https://www.wikidata.org/wiki/Q652707","display_name":"Lipschitz continuity","level":2,"score":0.4864000082015991},{"id":"https://openalex.org/C101597101","wikidata":"https://www.wikidata.org/wiki/Q2555655","display_name":"Minkowski\u2013Bouligand dimension","level":4,"score":0.4499000012874603},{"id":"https://openalex.org/C191399826","wikidata":"https://www.wikidata.org/wiki/Q326908","display_name":"Hausdorff space","level":2,"score":0.43959999084472656},{"id":"https://openalex.org/C115311070","wikidata":"https://www.wikidata.org/wiki/Q5347255","display_name":"Effective dimension","level":3,"score":0.4374000132083893},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.43220001459121704},{"id":"https://openalex.org/C2780388253","wikidata":"https://www.wikidata.org/wiki/Q5421508","display_name":"Exponent","level":2,"score":0.424699991941452},{"id":"https://openalex.org/C186450821","wikidata":"https://www.wikidata.org/wiki/Q17295","display_name":"Euclidean space","level":2,"score":0.42080000042915344},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4092999994754791},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.37929999828338623},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.37560001015663147},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.3596999943256378},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.35670000314712524},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.3449000120162964},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3199999928474426},{"id":"https://openalex.org/C133905733","wikidata":"https://www.wikidata.org/wiki/Q2629238","display_name":"Multifractal system","level":3,"score":0.2849000096321106},{"id":"https://openalex.org/C17327563","wikidata":"https://www.wikidata.org/wiki/Q1991405","display_name":"Nowhere dense set","level":3,"score":0.27239999175071716},{"id":"https://openalex.org/C40636538","wikidata":"https://www.wikidata.org/wiki/Q81392","display_name":"Fractal","level":2,"score":0.2678000032901764},{"id":"https://openalex.org/C164953516","wikidata":"https://www.wikidata.org/wiki/Q320357","display_name":"Closed set","level":2,"score":0.26350000500679016},{"id":"https://openalex.org/C66844729","wikidata":"https://www.wikidata.org/wiki/Q841816","display_name":"Inductive dimension","level":5,"score":0.25839999318122864},{"id":"https://openalex.org/C3019722297","wikidata":"https://www.wikidata.org/wiki/Q4440864","display_name":"High dimensional","level":2,"score":0.25110000371932983},{"id":"https://openalex.org/C188521023","wikidata":"https://www.wikidata.org/wiki/Q6795644","display_name":"Maximal set","level":3,"score":0.2506999969482422},{"id":"https://openalex.org/C68693459","wikidata":"https://www.wikidata.org/wiki/Q657586","display_name":"Hyperplane","level":2,"score":0.25060001015663147}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.04395","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.04395","pdf_url":"https://arxiv.org/pdf/2610.04395","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.2610.04395","doi":"https://doi.org/10.48550/arxiv.2610.04395","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.04395","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:2610.04395","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.04395","pdf_url":"https://arxiv.org/pdf/2610.04395","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/G1533368387","display_name":null,"funder_award_id":"12571096","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G4709341275","display_name":null,"funder_award_id":"12171107","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7220693539.pdf","grobid_xml":"https://content.openalex.org/works/W7220693539.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W2959356242","https://openalex.org/W4221158913","https://openalex.org/W7154728502","https://openalex.org/W7221175072","https://openalex.org/W2777408550","https://openalex.org/W7222006083","https://openalex.org/W4395664697","https://openalex.org/W4283463264","https://openalex.org/W6891996705","https://openalex.org/W2319639341"],"abstract_inverted_index":{"We":[0,130],"introduce":[1],"lower":[2,126],"and":[3,32,65,84,103,149,154,160],"upper":[4,110,117],"macroscopic":[5,39,93,111],"intermediate":[6,40,112,119],"dimensions":[7,41,64,91,134],"of":[8,11,42,87,135],"unbounded":[9,28],"subsets":[10],"Euclidean":[12],"space":[13],"by":[14],"combining":[15],"covering":[16],"contributions":[17],"from":[18],"all":[19],"dyadic":[20],"annuli":[21],"beyond":[22],"a":[23,72],"given":[24],"scale.":[25],"For":[26],"every":[27,52,66],"Borel":[29],"set":[30,60],"$E\\subset\\R^d$":[31],"$m\\in":[33],"\\{1,\\ldots,d-1\\}$,":[34],"we":[35],"prove":[36],"that":[37],"the":[38,45,116,125,132,142,147],"$P_VE$":[43],"equal":[44],"corresponding":[46],"intrinsic":[47],"capacity":[48],"profiles":[49],"for":[50,62,81],"almost":[51],"$m$-dimensional":[53],"linear":[54],"subspaces":[55],"$V\\subset":[56],"\\R^d$.":[57],"One":[58],"full-measure":[59],"works":[61],"both":[63],"parameter":[67,97],"$\u03b8\\in[0,1]$.":[68],"The":[69,90,109],"proof":[70],"combines":[71],"covering--capacity":[73],"comparison":[74],"with":[75,115],"an":[76],"exponent":[77],"gap,":[78],"fractional-moment":[79],"estimates":[80],"projected":[82,133],"coverings,":[83],"capacity-weighted":[85],"retention":[86],"source":[88],"shells.":[89],"recover":[92],"Hausdorff":[94],"dimension":[95,113,120],"at":[96,106],"zero,":[98],"are":[99,104,152],"invariant":[100],"under":[101],"quasi-isometries,":[102],"continuous":[105],"positive":[107],"parameters.":[108],"agrees":[114],"discrete":[118,137],"introduced":[121],"in":[122],"\\cite{LXZ},":[123],"whereas":[124],"one":[127],"can":[128],"differ.":[129],"compute":[131],"bounded-base":[136],"digit":[138,150],"Moran":[139],"sets,":[140],"including":[141],"self-similar":[143],"case":[144],"obtained":[145],"when":[146],"bases":[148],"sets":[151],"constant,":[153],"give":[155],"applications":[156],"to":[157],"Lipschitz":[158],"graphs":[159],"arithmetic":[161],"sumsets.":[162]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-10-07T00:00:00"}
