{"id":"https://openalex.org/W7204071805","doi":"https://doi.org/10.48550/arxiv.2608.21047","title":"On the horizon problem over the product of self-similar fractals","display_name":"On the horizon problem over the product of self-similar fractals","publication_year":2026,"publication_date":"2026-08-21","ids":{"openalex":"https://openalex.org/W7204071805","doi":"https://doi.org/10.48550/arxiv.2608.21047"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2608.21047","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.21047","pdf_url":"https://arxiv.org/pdf/2608.21047","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":"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/2608.21047","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5051570850","display_name":"Gurubachan","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I154549908","display_name":"Indian Institute of Technology Jodhpur","ror":"https://ror.org/03yacj906","country_code":"IN","type":"education","lineage":["https://openalex.org/I154549908"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Gurubachan","raw_affiliation_strings":["Department of Mathematics Indian Institute of Technology Jodhpur Jodhpur 342030 , India"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics Indian Institute of Technology Jodhpur Jodhpur 342030 , India","institution_ids":["https://openalex.org/I154549908"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5089723695","display_name":"S. Verma","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I26072440","display_name":"Indian Institute of Information Technology Allahabad","ror":"https://ror.org/03rgjt374","country_code":"IN","type":"education","lineage":["https://openalex.org/I26072440"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Verma, S.","raw_affiliation_strings":["Department of Applied Sciences Indian Institute of Information Technology Allahabad Prayagraj 211015 , India"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Applied Sciences Indian Institute of Information Technology Allahabad Prayagraj 211015 , India","institution_ids":["https://openalex.org/I26072440"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5048966120","display_name":"V. V. M. S. Chandramouli","orcid":"https://orcid.org/0000-0002-6331-2218","observed_orcids":["https://orcid.org/0000-0002-6331-2218"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Chandramouli, V. V. M. S.","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"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":0.9980000257492065,"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":0.9980000257492065,"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":0.0005000000237487257,"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/T12516","display_name":"Advanced Mathematical Theories and Applications","score":9.999999747378752e-05,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/horizon-problem","display_name":"horizon problem","score":0.7950000166893005},{"id":"https://openalex.org/keywords/box-counting-dimension","display_name":"box-counting dimension","score":0.8629999756813049},{"id":"https://openalex.org/keywords/self-similar-fractals","display_name":"self-similar fractals","score":0.9710000157356262},{"id":"https://openalex.org/keywords/iterated-function-systems","display_name":"iterated function systems","score":0.9570000171661377},{"id":"https://openalex.org/keywords/open-set-condition","display_name":"open set condition","score":0.9980000257492065},{"id":"https://openalex.org/keywords/fractal-geometry","display_name":"fractal geometry","score":0.7120000123977661}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6820999979972839},{"id":"https://openalex.org/C40636538","wikidata":"https://www.wikidata.org/wiki/Q81392","display_name":"Fractal","level":2,"score":0.6444000005722046},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.6396999955177307},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.5827999711036682},{"id":"https://openalex.org/C101597101","wikidata":"https://www.wikidata.org/wiki/Q2555655","display_name":"Minkowski\u2013Bouligand dimension","level":4,"score":0.5080999732017517},{"id":"https://openalex.org/C159176650","wikidata":"https://www.wikidata.org/wiki/Q43261","display_name":"Horizon","level":2,"score":0.478300005197525},{"id":"https://openalex.org/C115311070","wikidata":"https://www.wikidata.org/wiki/Q5347255","display_name":"Effective dimension","level":3,"score":0.4747999906539917},{"id":"https://openalex.org/C99008458","wikidata":"https://www.wikidata.org/wiki/Q4951583","display_name":"Box counting","level":5,"score":0.46720001101493835},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.44209998846054077},{"id":"https://openalex.org/C28235433","wikidata":"https://www.wikidata.org/wiki/Q5277259","display_name":"Dimension function","level":3,"score":0.4417000114917755},{"id":"https://openalex.org/C26546657","wikidata":"https://www.wikidata.org/wiki/Q1412452","display_name":"Fractal dimension","level":3,"score":0.4399000108242035},{"id":"https://openalex.org/C189950617","wikidata":"https://www.wikidata.org/wiki/Q937228","display_name":"Property (philosophy)","level":2,"score":0.438400000333786},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.41839998960494995},{"id":"https://openalex.org/C138888516","wikidata":"https://www.wikidata.org/wiki/Q2266843","display_name":"Infinite product","level":2,"score":0.398499995470047},{"id":"https://openalex.org/C60396315","wikidata":"https://www.wikidata.org/wiki/Q5477489","display_name":"Fractal dimension on networks","level":5,"score":0.3788999915122986},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.3686999976634979},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3440000116825104},{"id":"https://openalex.org/C194198291","wikidata":"https://www.wikidata.org/wiki/Q565186","display_name":"Hausdorff dimension","level":2,"score":0.3375000059604645},{"id":"https://openalex.org/C80661125","wikidata":"https://www.wikidata.org/wiki/Q5404299","display_name":"Packing dimension","level":5,"score":0.33219999074935913},{"id":"https://openalex.org/C133905733","wikidata":"https://www.wikidata.org/wiki/Q2629238","display_name":"Multifractal system","level":3,"score":0.30300000309944153},{"id":"https://openalex.org/C2778199601","wikidata":"https://www.wikidata.org/wiki/Q5165476","display_name":"Continuous function (set theory)","level":3,"score":0.3005000054836273},{"id":"https://openalex.org/C123222754","wikidata":"https://www.wikidata.org/wiki/Q1070472","display_name":"Product topology","level":2,"score":0.2863999903202057},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.2802000045776367},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.27140000462532043}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2608.21047","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.21047","pdf_url":"https://arxiv.org/pdf/2608.21047","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2608.21047","doi":"https://doi.org/10.48550/arxiv.2608.21047","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.21047","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:2608.21047","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.21047","pdf_url":"https://arxiv.org/pdf/2608.21047","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320322724","display_name":"Ministry of Education, India","ror":"https://ror.org/048xjjh50"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7204071805.pdf","grobid_xml":"https://content.openalex.org/works/W7204071805.grobid-xml"},"referenced_works_count":19,"referenced_works":["https://openalex.org/W1515025867","https://openalex.org/W1995246440","https://openalex.org/W2082258496","https://openalex.org/W2089668757","https://openalex.org/W2090331392","https://openalex.org/W2560630848","https://openalex.org/W2900804084","https://openalex.org/W3196250463","https://openalex.org/W4238652859","https://openalex.org/W4249775755","https://openalex.org/W4290635493","https://openalex.org/W4383554669","https://openalex.org/W4388093908","https://openalex.org/W4391053216","https://openalex.org/W4402283506","https://openalex.org/W4402309876","https://openalex.org/W4415008521","https://openalex.org/W4416061396","https://openalex.org/W4416292081"],"related_works":["https://openalex.org/W3101882843","https://openalex.org/W2089668757","https://openalex.org/W4323356207","https://openalex.org/W1649039880","https://openalex.org/W4394601734","https://openalex.org/W2807576821","https://openalex.org/W2030307273","https://openalex.org/W4376253928","https://openalex.org/W4388333347","https://openalex.org/W4395009956"],"abstract_inverted_index":{"In":[0],"this":[1],"article,":[2],"we":[3,41,100],"first":[4],"investigate":[5],"the":[6,11,23,34,43,59,65,69,73,91,95,106,109],"box":[7,66,70,92],"dimension":[8,71,93],"results":[9],"for":[10],"graphs":[12],"of":[13,45,72,80,94,112],"continuous":[14,52],"functions":[15],"defined":[16,49],"on":[17,54,108],"$\\mathfrak{F\\times":[18,55],"F}$,":[19,56],"where":[20,88],"$\\mathfrak{F}$":[21,113],"is":[22,76,90],"self-similar":[24,96],"fractal":[25,97],"generated":[26],"by":[27,50],"an":[28,102],"iterated":[29],"function":[30,53],"system":[31],"(IFS)":[32],"exhibiting":[33],"open":[35],"set":[36],"condition":[37],"(OSC).":[38],"Following":[39],"this,":[40],"manifest":[42],"existence":[44],"a":[46,51,85],"prevalent":[47,74],"surface,":[48],"that":[57,79,105],"satisfies":[58],"horizon":[60,82],"property":[61],"in":[62],"relation":[63],"to":[64,84],"dimension,":[67],"i.e.,":[68],"surface":[75],"greater":[77],"than":[78],"its":[81],"up":[83],"constant":[86],"$\u03b7$,":[87],"$\u03b7$":[89],"$\\mathfrak{F}$.":[98],"Finally,":[99],"provide":[101],"example":[103],"consolidating":[104],"OSC":[107],"underlying":[110],"IFS":[111],"cannot":[114],"be":[115],"dropped.":[116]},"counts_by_year":[],"updated_date":"2026-11-08T12:12:28.789561","created_date":"2026-08-25T00:00:00"}