{"id":"https://openalex.org/W7214628286","doi":"https://doi.org/10.48550/arxiv.2609.30680","title":"A Log-Star Comparison Between Expectation Threshold and Fractional Expectation Threshold","display_name":"A Log-Star Comparison Between Expectation Threshold and Fractional Expectation Threshold","publication_year":2026,"publication_date":"2026-09-25","ids":{"openalex":"https://openalex.org/W7214628286","doi":"https://doi.org/10.48550/arxiv.2609.30680"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.30680","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.30680","pdf_url":"https://arxiv.org/pdf/2609.30680","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/2609.30680","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5153351398","display_name":"Xuan Fang","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Fang, Xuan","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5100417408","display_name":"Tianyu Wang","orcid":"https://orcid.org/0000-0001-6206-8359","observed_orcids":["https://orcid.org/0000-0001-6206-8359","https://orcid.org/0000-0003-3990-1108"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Wang, Tianyu","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":0.9897000193595886,"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":0.9897000193595886,"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":0.0038999998942017555,"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/T13500","display_name":"Probability and Statistical Research","score":0.0006000000284984708,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"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/iterated-logarithm","display_name":"iterated logarithm","score":0.8140000104904175},{"id":"https://openalex.org/keywords/extremal-set-theory","display_name":"extremal set theory","score":0.75},{"id":"https://openalex.org/keywords/vc-dimension","display_name":"VC dimension","score":0.8899999856948853}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7124000191688538},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.5166000127792358},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.4952999949455261},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.43230000138282776},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.35260000824928284},{"id":"https://openalex.org/C202632270","wikidata":"https://www.wikidata.org/wiki/Q7798106","display_name":"Threshold model","level":2,"score":0.32899999618530273},{"id":"https://openalex.org/C3019361169","wikidata":"https://www.wikidata.org/wiki/Q2609467","display_name":"Detection threshold","level":2,"score":0.3084999918937683},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.30250000953674316},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.29499998688697815},{"id":"https://openalex.org/C2778067643","wikidata":"https://www.wikidata.org/wiki/Q166507","display_name":"Interval (graph theory)","level":2,"score":0.27630001306533813},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.2759999930858612},{"id":"https://openalex.org/C149441793","wikidata":"https://www.wikidata.org/wiki/Q200726","display_name":"Probability distribution","level":2,"score":0.2685999870300293},{"id":"https://openalex.org/C8272713","wikidata":"https://www.wikidata.org/wiki/Q176737","display_name":"Stochastic process","level":2,"score":0.2667999863624573},{"id":"https://openalex.org/C99498987","wikidata":"https://www.wikidata.org/wiki/Q2210247","display_name":"Noise (video)","level":3,"score":0.2653000056743622},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.25940001010894775},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.25699999928474426},{"id":"https://openalex.org/C139676723","wikidata":"https://www.wikidata.org/wiki/Q1193832","display_name":"Sign (mathematics)","level":2,"score":0.25209999084472656}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.30680","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.30680","pdf_url":"https://arxiv.org/pdf/2609.30680","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.2609.30680","doi":"https://doi.org/10.48550/arxiv.2609.30680","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.30680","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:2609.30680","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.30680","pdf_url":"https://arxiv.org/pdf/2609.30680","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":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214628286.pdf","grobid_xml":"https://content.openalex.org/works/W7214628286.grobid-xml"},"referenced_works_count":4,"referenced_works":["https://openalex.org/W4405033683","https://openalex.org/W4405090808","https://openalex.org/W7212087038","https://openalex.org/W7213280207"],"related_works":["https://openalex.org/W7213280207","https://openalex.org/W4414110457","https://openalex.org/W4416046449","https://openalex.org/W2982592370","https://openalex.org/W4405033683","https://openalex.org/W3165408995","https://openalex.org/W4385899778","https://openalex.org/W7214364543","https://openalex.org/W4405802694","https://openalex.org/W1922270879"],"abstract_inverted_index":{"We":[0],"proved":[1],"a":[2,26,65,102],"log-star":[3],"comparison":[4],"between":[5],"the":[6,12,45,59,91,105,114],"expectation":[7,14],"threshold":[8,15],"$q(\\mathcal":[9],"F)$":[10,17,89],"and":[11,87,109],"fractional":[13,108],"$q_f(\\mathcal":[16],"for":[18,125],"any":[19],"nontrivial":[20],"increasing":[21],"family":[22],"$\\mathcal":[23],"F$":[24],"on":[25],"finite":[27],"ground":[28],"set":[29],"$V$":[30],"of":[31,48,93,107],"size":[32],"$|V|=N$.":[33],"Specifically,":[34],"we":[35],"show":[36],"that":[37,70],"$$q_f(\\mathcal":[38],"F)\\le64\\log_2^*(N+2)\\,q(\\mathcal":[39],"F),$$":[40],"where":[41,83],"$\\log_2^*":[42],"x$":[43],"is":[44],"iterated":[46],"logarithm":[47],"base-2.":[49],"Using":[50],"similiar":[51],"methods,":[52],"one":[53],"can":[54],"extend":[55],"this":[56,117,123],"result":[57,103],"to":[58],"following":[60],"stronger":[61],"form:":[62],"there":[63],"exists":[64],"universal":[66],"constant":[67],"$C>0$":[68],"such":[69],"\\[":[71],"q_f(\\mathcal":[72,77],"F)\\le":[73,78],"Cq(\\mathcal":[74,79],"F)\\log_2^*(l(\\mathcal":[75],"F)),\\quad":[76],"F)\\log_2^*(\\operatorname{VC}(\\min\\mathcal":[80],"F)),":[81],"\\]":[82],"$l(\\mathcal":[84],"F):=\\max\\{2,\\max_{S\\in\\min\\mathcal":[85],"F}|S|\\}$":[86],"$\\operatorname{VC}(\\min\\mathcal":[88],"denotes":[90],"VC-dimension":[92],"$\\min\\mathcal":[94],"F$.":[95],"On":[96],"October":[97],"7,":[98],"2026,":[99],"OpenAI":[100],"released":[101],"proving":[104],"equivalence":[106],"integral":[110],"thresholds,":[111],"which":[112],"supersedes":[113],"findings":[115],"in":[116],"note.":[118],"The":[119],"authors":[120],"are":[121],"keeping":[122],"note":[124],"historical":[126],"reference.":[127]},"counts_by_year":[],"updated_date":"2026-11-08T12:12:28.789561","created_date":"2026-09-29T00:00:00"}
