{"id":"https://openalex.org/W7214328249","doi":"https://doi.org/10.48550/arxiv.2609.29301","title":"Absolute continuity of projections and dimension conservation for self-similar sets and measures","display_name":"Absolute continuity of projections and dimension conservation for self-similar sets and measures","publication_year":2026,"publication_date":"2026-09-24","ids":{"openalex":"https://openalex.org/W7214328249","doi":"https://doi.org/10.48550/arxiv.2609.29301"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.29301","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.29301","pdf_url":"https://arxiv.org/pdf/2609.29301","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/2609.29301","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5031302353","display_name":"Xiong Jin","orcid":"https://orcid.org/0000-0002-4468-3187","observed_orcids":["https://orcid.org/0000-0002-4468-3187"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Jin, Xiong","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5022780659","display_name":"Tuomas Sahlsten","orcid":"https://orcid.org/0000-0001-5625-5481","observed_orcids":["https://orcid.org/0000-0001-5625-5481"]},"institutions":[{"id":"https://openalex.org/I133731052","display_name":"University of Helsinki","ror":"https://ror.org/040af2s02","country_code":"FI","type":"education","lineage":["https://openalex.org/I133731052"]},{"id":"https://openalex.org/I28407311","display_name":"University of Manchester","ror":"https://ror.org/027m9bs27","country_code":"GB","type":"education","lineage":["https://openalex.org/I28407311"]}],"countries":["FI","GB"],"is_corresponding":false,"raw_author_name":"Sahlsten, Tuomas","raw_affiliation_strings":["Department of Mathematics , The University of Manchester , Oxford Road , Manchester M13 9PL , United Kingdom","Department of Mathematics and Statistics , University of Helsinki , P.O. Box 68 , FI-00014"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , The University of Manchester , Oxford Road , Manchester M13 9PL , United Kingdom","institution_ids":["https://openalex.org/I28407311"]},{"raw_affiliation_string":"Department of Mathematics and Statistics , University of Helsinki , P.O. Box 68 , FI-00014","institution_ids":["https://openalex.org/I133731052"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":1,"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":0.9984999895095825,"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":0.9984999895095825,"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":0.00039999998989515007,"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/T11152","display_name":"Stochastic processes and statistical mechanics","score":9.999999747378752e-05,"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/self-similar-sets","display_name":"self-similar sets","score":0.9399999976158142},{"id":"https://openalex.org/keywords/self-similar-measures","display_name":"self-similar measures","score":1.0},{"id":"https://openalex.org/keywords/hausdorff-dimension","display_name":"Hausdorff dimension","score":0.7820000052452087},{"id":"https://openalex.org/keywords/conditional-measures","display_name":"conditional measures","score":0.7570000290870667},{"id":"https://openalex.org/keywords/strong-separation-condition","display_name":"strong separation condition","score":0.5979999899864197},{"id":"https://openalex.org/keywords/gaussian-blur","display_name":"Gaussian smoothing","score":0.8679999709129333},{"id":"https://openalex.org/keywords/iterated-function-systems","display_name":"iterated function systems","score":0.7289999723434448}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8921999931335449},{"id":"https://openalex.org/C2777105136","wikidata":"https://www.wikidata.org/wiki/Q827230","display_name":"Lebesgue measure","level":3,"score":0.7023000121116638},{"id":"https://openalex.org/C194198291","wikidata":"https://www.wikidata.org/wiki/Q565186","display_name":"Hausdorff dimension","level":2,"score":0.6398000121116638},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.6312000155448914},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.5823000073432922},{"id":"https://openalex.org/C118733216","wikidata":"https://www.wikidata.org/wiki/Q332504","display_name":"Absolute continuity","level":2,"score":0.5317999720573425},{"id":"https://openalex.org/C3770464","wikidata":"https://www.wikidata.org/wiki/Q775963","display_name":"Smoothing","level":2,"score":0.516700029373169},{"id":"https://openalex.org/C14158598","wikidata":"https://www.wikidata.org/wiki/Q853831","display_name":"Lebesgue integration","level":2,"score":0.510200023651123},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.4246000051498413},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.4185999929904938},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4153999984264374},{"id":"https://openalex.org/C115311070","wikidata":"https://www.wikidata.org/wiki/Q5347255","display_name":"Effective dimension","level":3,"score":0.39089998602867126},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3837999999523163},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3808000087738037},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.3601999878883362},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.3368000090122223},{"id":"https://openalex.org/C203024314","wikidata":"https://www.wikidata.org/wiki/Q1365258","display_name":"Fourier analysis","level":3,"score":0.33660000562667847},{"id":"https://openalex.org/C12362212","wikidata":"https://www.wikidata.org/wiki/Q728435","display_name":"Linear subspace","level":2,"score":0.3246000111103058},{"id":"https://openalex.org/C73225184","wikidata":"https://www.wikidata.org/wiki/Q1591095","display_name":"Hausdorff measure","level":3,"score":0.31859999895095825},{"id":"https://openalex.org/C80661125","wikidata":"https://www.wikidata.org/wiki/Q5404299","display_name":"Packing dimension","level":5,"score":0.31439998745918274},{"id":"https://openalex.org/C74050887","wikidata":"https://www.wikidata.org/wiki/Q848368","display_name":"Rotation (mathematics)","level":2,"score":0.3111000061035156},{"id":"https://openalex.org/C102519508","wikidata":"https://www.wikidata.org/wiki/Q6520159","display_name":"Fourier transform","level":2,"score":0.3052000105381012},{"id":"https://openalex.org/C21031990","wikidata":"https://www.wikidata.org/wiki/Q355020","display_name":"Probability measure","level":2,"score":0.3018999993801117},{"id":"https://openalex.org/C75432250","wikidata":"https://www.wikidata.org/wiki/Q29654355","display_name":"Real line","level":2,"score":0.3018999993801117},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.30149999260902405},{"id":"https://openalex.org/C191399826","wikidata":"https://www.wikidata.org/wiki/Q326908","display_name":"Hausdorff space","level":2,"score":0.29010000824928284},{"id":"https://openalex.org/C101597101","wikidata":"https://www.wikidata.org/wiki/Q2555655","display_name":"Minkowski\u2013Bouligand dimension","level":4,"score":0.2874000072479248},{"id":"https://openalex.org/C17327563","wikidata":"https://www.wikidata.org/wiki/Q1991405","display_name":"Nowhere dense set","level":3,"score":0.2824000120162964},{"id":"https://openalex.org/C26546657","wikidata":"https://www.wikidata.org/wiki/Q1412452","display_name":"Fractal dimension","level":3,"score":0.2815999984741211},{"id":"https://openalex.org/C141898687","wikidata":"https://www.wikidata.org/wiki/Q1501997","display_name":"Hausdorff distance","level":2,"score":0.2745000123977661},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.2669999897480011},{"id":"https://openalex.org/C150591576","wikidata":"https://www.wikidata.org/wiki/Q739925","display_name":"Family of sets","level":3,"score":0.263700008392334},{"id":"https://openalex.org/C17137986","wikidata":"https://www.wikidata.org/wiki/Q215067","display_name":"Orthogonality","level":2,"score":0.2603999972343445},{"id":"https://openalex.org/C23707678","wikidata":"https://www.wikidata.org/wiki/Q164262","display_name":"Lebesgue covering dimension","level":3,"score":0.2502000033855438}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.29301","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.29301","pdf_url":"https://arxiv.org/pdf/2609.29301","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.2609.29301","doi":"https://doi.org/10.48550/arxiv.2609.29301","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.29301","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:2609.29301","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.29301","pdf_url":"https://arxiv.org/pdf/2609.29301","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/W7214328249.pdf","grobid_xml":"https://content.openalex.org/works/W7214328249.grobid-xml"},"referenced_works_count":34,"referenced_works":["https://openalex.org/W1481054052","https://openalex.org/W1575083275","https://openalex.org/W1595889759","https://openalex.org/W1985551666","https://openalex.org/W2031808668","https://openalex.org/W2067849838","https://openalex.org/W2068585082","https://openalex.org/W2073997685","https://openalex.org/W2075651598","https://openalex.org/W2115798948","https://openalex.org/W2120045167","https://openalex.org/W2152491771","https://openalex.org/W2169450769","https://openalex.org/W2178311504","https://openalex.org/W2332552440","https://openalex.org/W2525870670","https://openalex.org/W2605295300","https://openalex.org/W2894302755","https://openalex.org/W2911612467","https://openalex.org/W2925113957","https://openalex.org/W2962754787","https://openalex.org/W2962852664","https://openalex.org/W3162141967","https://openalex.org/W4249775755","https://openalex.org/W4312882633","https://openalex.org/W4407245144","https://openalex.org/W4407245150","https://openalex.org/W4408981597","https://openalex.org/W4409563962","https://openalex.org/W4412458701","https://openalex.org/W4412617603","https://openalex.org/W7169551624","https://openalex.org/W7169762633","https://openalex.org/W7214024557"],"related_works":["https://openalex.org/W2894302755","https://openalex.org/W7095600190","https://openalex.org/W2075651598","https://openalex.org/W3098741874","https://openalex.org/W7211865548","https://openalex.org/W2904423372","https://openalex.org/W4297909692","https://openalex.org/W7169762633","https://openalex.org/W2208514259","https://openalex.org/W2536213605"],"abstract_inverted_index":{"Let":[0],"$A\\subset\\mathbb{R}^d$,":[1],"$d\\geq3$,":[2],"be":[3,116],"the":[4,65,69,88,97,122],"attractor":[5],"of":[6,16,74,85,87,90],"a":[7,13,48],"self-similar":[8,49],"IFS":[9,54],"whose":[10],"rotations":[11],"generate":[12],"dense":[14,112],"subgroup":[15],"$\\mathrm{SO}(d)$.":[17],"For":[18],"every":[19,42],"$1\\leq":[20],"k<\\dim_{\\mathrm{H}}":[21],"A$":[22],"and":[23,37,118],"$V\\in\\mathrm{Gr}(d,k)$,":[24],"we":[25],"prove":[26],"that":[27],"$\u03c0_VA$":[28],"has":[29],"positive":[30],"$k$-dimensional":[31],"Lebesgue":[32],"measure":[33,50],"uniformly":[34],"in":[35],"$V$,":[36],"$\\dim_{\\mathrm{H}}(A\\cap\u03c0_V^{-1}\\{y\\})=\\dim_{\\mathrm{H}}":[38],"A-k$":[39],"for":[40,51,57],"Lebesgue-almost":[41],"$y\\in\u03c0_VA$.":[43],"Moreover,":[44],"if":[45],"$\u03bc$":[46],"is":[47],"such":[52],"an":[53],"with":[55],"$I_t(\u03bc)<\\infty$":[56],"some":[58],"$kM\\}}f_V\\,d\\mathcal{L}_V^k\\lesssim":[59],"e^{-c(\\log":[60],"M)^{1/3}}$.":[61],"Under":[62],"strong":[63],"separation,":[64],"conditional":[66],"measures":[67],"on":[68],"fibres":[70],"are":[71],"exact":[72],"dimensional":[73],"dimension":[75],"$\\dim_{\\mathrm{H}}\u03bc-k$.":[76],"The":[77],"proof":[78],"uses":[79],"self-similarity":[80],"to":[81,115,120],"iterate":[82],"$L^1$":[83,98],"norms":[84],"differences":[86],"Gaussian-smoothing":[89],"$\u03bc_V$":[91],"between":[92],"scales.":[93],"Unlike":[94],"$L^q$,":[95],"$q>1$,":[96],"norm":[99],"does":[100],"not":[101],"grow":[102],"under":[103,111],"rescaling,":[104],"allowing":[105],"Varj\u00fa's":[106],"$L^2$":[107],"weak":[108],"gap":[109,127],"theorem":[110],"support":[113],"condition":[114],"used,":[117],"avoiding":[119],"use":[121],"much":[123],"stronger":[124],"uniform":[125],"spectral":[126],"assumption.":[128]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-08T12:12:28.789561","created_date":"2026-09-26T00:00:00"}