{"id":"https://openalex.org/W7214736746","doi":"https://doi.org/10.48550/arxiv.2609.33526","title":"Explicit Fractalizers: Prime-Order Paley Graphs and other Cayley Graphs","display_name":"Explicit Fractalizers: Prime-Order Paley Graphs and other Cayley Graphs","publication_year":2026,"publication_date":"2026-09-27","ids":{"openalex":"https://openalex.org/W7214736746","doi":"https://doi.org/10.48550/arxiv.2609.33526"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.33526","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.33526","pdf_url":"https://arxiv.org/pdf/2609.33526","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/2609.33526","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5093497303","display_name":"Aldo Kiem","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Kiem, Aldo","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5100911449","display_name":"Fan Wei","orcid":"https://orcid.org/0009-0003-9123-0396","observed_orcids":["https://orcid.org/0009-0003-9123-0396"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Wei, Fan","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/T11329","display_name":"Limits and Structures in Graph Theory","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/T11329","display_name":"Limits and Structures in Graph Theory","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.0,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/T10374","display_name":"Advanced Graph Theory Research","score":0.0,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/inducibility","display_name":"inducibility","score":0.8989999890327454},{"id":"https://openalex.org/keywords/induced-subgraphs","display_name":"induced subgraphs","score":0.7730000019073486},{"id":"https://openalex.org/keywords/cayley-graphs","display_name":"Cayley graphs","score":0.9480000138282776},{"id":"https://openalex.org/keywords/extremal-graphs","display_name":"extremal graphs","score":0.8619999885559082},{"id":"https://openalex.org/keywords/explicit-constructions","display_name":"explicit constructions","score":0.8399999737739563},{"id":"https://openalex.org/keywords/extremal-graph-theory","display_name":"extremal graph theory","score":0.5870000123977661},{"id":"https://openalex.org/keywords/flag-algebras","display_name":"flag algebras","score":0.7599999904632568}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7907999753952026},{"id":"https://openalex.org/C120204988","wikidata":"https://www.wikidata.org/wiki/Q859174","display_name":"Cayley graph","level":3,"score":0.6951000094413757},{"id":"https://openalex.org/C140479938","wikidata":"https://www.wikidata.org/wiki/Q5254619","display_name":"Iterated function","level":2,"score":0.6606000065803528},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6453999876976013},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5856999754905701},{"id":"https://openalex.org/C47458327","wikidata":"https://www.wikidata.org/wiki/Q910404","display_name":"Random graph","level":3,"score":0.536899983882904},{"id":"https://openalex.org/C549619432","wikidata":"https://www.wikidata.org/wiki/Q5422299","display_name":"Extremal combinatorics","level":2,"score":0.48890000581741333},{"id":"https://openalex.org/C136170076","wikidata":"https://www.wikidata.org/wiki/Q181296","display_name":"Abelian group","level":2,"score":0.4361000061035156},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.414900004863739},{"id":"https://openalex.org/C160446614","wikidata":"https://www.wikidata.org/wiki/Q1322892","display_name":"Chordal graph","level":3,"score":0.414900004863739},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.4049000144004822},{"id":"https://openalex.org/C187407849","wikidata":"https://www.wikidata.org/wiki/Q6889712","display_name":"Modular decomposition","level":5,"score":0.38999998569488525},{"id":"https://openalex.org/C4255713","wikidata":"https://www.wikidata.org/wiki/Q7291988","display_name":"Random regular graph","level":5,"score":0.38690000772476196},{"id":"https://openalex.org/C113203676","wikidata":"https://www.wikidata.org/wiki/Q220997","display_name":"Vertex-transitive graph","level":5,"score":0.3815999925136566},{"id":"https://openalex.org/C49937458","wikidata":"https://www.wikidata.org/wiki/Q2599292","display_name":"Probabilistic logic","level":2,"score":0.3619999885559082},{"id":"https://openalex.org/C14201764","wikidata":"https://www.wikidata.org/wiki/Q1205074","display_name":"Symmetric graph","level":5,"score":0.36079999804496765},{"id":"https://openalex.org/C74133993","wikidata":"https://www.wikidata.org/wiki/Q3115472","display_name":"Indifference graph","level":3,"score":0.35010001063346863},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.34540000557899475},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.3449000120162964},{"id":"https://openalex.org/C68103157","wikidata":"https://www.wikidata.org/wiki/Q569347","display_name":"Graph product","level":5,"score":0.3228999972343445},{"id":"https://openalex.org/C59824394","wikidata":"https://www.wikidata.org/wiki/Q5141281","display_name":"Cograph","level":5,"score":0.3061000108718872},{"id":"https://openalex.org/C189950617","wikidata":"https://www.wikidata.org/wiki/Q937228","display_name":"Property (philosophy)","level":2,"score":0.30329999327659607},{"id":"https://openalex.org/C102192266","wikidata":"https://www.wikidata.org/wiki/Q4545823","display_name":"1-planar graph","level":4,"score":0.3009999990463257},{"id":"https://openalex.org/C203776342","wikidata":"https://www.wikidata.org/wiki/Q1378376","display_name":"Line graph","level":3,"score":0.29809999465942383},{"id":"https://openalex.org/C182419690","wikidata":"https://www.wikidata.org/wiki/Q205464","display_name":"Algebraic structure","level":2,"score":0.28519999980926514},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.28049999475479126},{"id":"https://openalex.org/C54643580","wikidata":"https://www.wikidata.org/wiki/Q1005603","display_name":"Algebraic combinatorics","level":2,"score":0.27869999408721924},{"id":"https://openalex.org/C43517604","wikidata":"https://www.wikidata.org/wiki/Q7144893","display_name":"Pathwidth","level":4,"score":0.2766000032424927},{"id":"https://openalex.org/C146380142","wikidata":"https://www.wikidata.org/wiki/Q1137726","display_name":"Directed graph","level":2,"score":0.2745000123977661},{"id":"https://openalex.org/C43413943","wikidata":"https://www.wikidata.org/wiki/Q4927193","display_name":"Block graph","level":5,"score":0.2687000036239624},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.26649999618530273},{"id":"https://openalex.org/C64339825","wikidata":"https://www.wikidata.org/wiki/Q722659","display_name":"Graph property","level":5,"score":0.25429999828338623},{"id":"https://openalex.org/C2780992000","wikidata":"https://www.wikidata.org/wiki/Q17016113","display_name":"Generator (circuit theory)","level":3,"score":0.2522999942302704}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.33526","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.33526","pdf_url":"https://arxiv.org/pdf/2609.33526","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.2609.33526","doi":"https://doi.org/10.48550/arxiv.2609.33526","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.33526","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:2609.33526","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.33526","pdf_url":"https://arxiv.org/pdf/2609.33526","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/G1372115314","display_name":null,"funder_award_id":"EXC-2046","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G2033548659","display_name":null,"funder_award_id":"EXC\u20102046/1","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G7725451807","display_name":"EXC 2046: MATH+: Berlin Mathematics Research Center","funder_award_id":"390685689","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"}],"funders":[{"id":"https://openalex.org/F4320320879","display_name":"Deutsche Forschungsgemeinschaft","ror":"https://ror.org/018mejw64"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214736746.pdf","grobid_xml":"https://content.openalex.org/works/W7214736746.grobid-xml"},"referenced_works_count":3,"referenced_works":["https://openalex.org/W3135658926","https://openalex.org/W4416909221","https://openalex.org/W7163656931"],"related_works":["https://openalex.org/W3160710271","https://openalex.org/W3135658926","https://openalex.org/W4361270144","https://openalex.org/W3108706109","https://openalex.org/W3003459502","https://openalex.org/W4404988730","https://openalex.org/W7206163999","https://openalex.org/W2407999126","https://openalex.org/W2781654496","https://openalex.org/W4416909221"],"abstract_inverted_index":{"The":[0],"inducibility":[1],"problem":[2],"asks":[3],"for":[4,40,66,183],"the":[5,48,75,93,151,169,185,194,201,206],"maximum":[6],"number":[7,21,76],"of":[8,11,22,30,77,80,87,155,179,210],"induced":[9,78],"copies":[10,79],"a":[12,19,63,83,143,174,189],"fixed":[13],"graph":[14,59,73,141],"among":[15],"all":[16,37],"graphs":[17,39,112,117],"with":[18,120],"prescribed":[20],"vertices.":[23],"Inducibility":[24],"has":[25],"been":[26],"an":[27],"active":[28],"area":[29],"research":[31],"in":[32,96,193],"extremal":[33,38,157],"combinatorics,":[34],"but":[35],"determining":[36],"explicitly":[41],"defined":[42],"patterns":[43,149],"remains":[44],"challenging,":[45],"particularly":[46],"when":[47],"description":[49],"is":[50,61,82,142],"required":[51],"to":[52,123],"hold":[53],"at":[54,102,159,205],"every":[55,67,71,136,156,160],"host":[56,161],"order.":[57],"A":[58],"$H$":[60,81],"called":[62],"fractalizer":[64,186],"if,":[65],"positive":[68],"integer":[69],"$n$,":[70],"$n$-vertex":[72],"maximizing":[74],"balanced":[84],"iterated":[85],"blow-up":[86],"$H$,":[88],"obtained":[89],"by":[90,101],"recursively":[91],"repeating":[92],"same":[94],"pattern":[95],"parts":[97],"whose":[98],"sizes":[99],"differ":[100],"most":[103],"one.":[104],"Previous":[105],"probabilistic":[106],"results":[107,199],"show":[108],"that":[109,135],"large":[110,138],"random":[111,114],"and":[113,163,215],"abelian":[115],"Cayley":[116,181],"are":[118],"fractalizers":[119],"probability":[121],"tending":[122],"one,":[124],"establishing":[125],"their":[126],"abundance":[127],"without":[128,164],"directly":[129],"providing":[130],"explicit":[131,176],"families.":[132],"We":[133,171],"prove":[134],"sufficiently":[137],"prime-order":[139],"Paley":[140,195],"fractalizer.":[144],"Thus":[145],"these":[146,198],"classical":[147],"arithmetic":[148],"determine":[150],"exact":[152],"recursive":[153],"structure":[154],"host,":[158],"order":[162],"any":[165],"algebraic":[166],"assumptions":[167],"on":[168],"host.":[170],"also":[172],"construct":[173],"second":[175],"infinite":[177],"family":[178],"nontrivial":[180],"fractalizers,":[182],"which":[184],"property":[187],"admits":[188],"simpler":[190],"proof":[191],"than":[192],"case.":[196],"Together,":[197],"resolve":[200],"explicit-construction":[202],"question":[203],"discussed":[204],"2025":[207],"American":[208],"Institute":[209],"Mathematics":[211],"workshop":[212],"\"Flag":[213],"Algebras":[214],"Extremal":[216],"Combinatorics.\"":[217]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-30T00:00:00"}