{"id":"https://openalex.org/W7208764988","doi":"https://doi.org/10.48550/arxiv.2608.24401","title":"Winning property of counterexamples to Uniform Littlewood's Conjecture","display_name":"Winning property of counterexamples to Uniform Littlewood's Conjecture","publication_year":2026,"publication_date":"2026-08-25","ids":{"openalex":"https://openalex.org/W7208764988","doi":"https://doi.org/10.48550/arxiv.2608.24401"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2608.24401","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.24401","pdf_url":"https://arxiv.org/pdf/2608.24401","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/2608.24401","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5088509387","display_name":"Vasiliy Neckrasov","orcid":"https://orcid.org/0009-0006-5203-3500","observed_orcids":["https://orcid.org/0009-0006-5203-3500"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Neckrasov, Vasiliy","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5110127669","display_name":"C.-Y. Wu","orcid":"https://orcid.org/0009-0005-6070-5007","observed_orcids":["https://orcid.org/0009-0005-6070-5007"]},"institutions":[{"id":"https://openalex.org/I4403928334","display_name":"Shanghai Institute for Mathematics and Interdisciplinary Sciences","ror":"https://ror.org/0489cd473","country_code":"CN","type":"government","lineage":["https://openalex.org/I4403928334"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Wu, Chengyang","raw_affiliation_strings":["Shanghai Institute for Mathematics and Interdisciplinary Sciences , Shanghai 200433 , China Research Institute of Intelligent Complex Systems ,"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Shanghai Institute for Mathematics and Interdisciplinary Sciences , Shanghai 200433 , China Research Institute of Intelligent Complex Systems ,","institution_ids":["https://openalex.org/I4403928334"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5150053292","display_name":"Bohan Yang","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I24943067","display_name":"Fudan University","ror":"https://ror.org/013q1eq08","country_code":"CN","type":"education","lineage":["https://openalex.org/I24943067"]},{"id":"https://openalex.org/I40347166","display_name":"University of Chicago","ror":"https://ror.org/024mw5h28","country_code":"US","type":"education","lineage":["https://openalex.org/I40347166"]}],"countries":["CN","US"],"is_corresponding":false,"raw_author_name":"Yang, Bohan","raw_affiliation_strings":["Department of Mathematics , University of Chicago , Chicago , Illinois , 60637 , U.S","Fudan University , Shanghai 200433 , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , University of Chicago , Chicago , Illinois , 60637 , U.S","institution_ids":["https://openalex.org/I40347166"]},{"raw_affiliation_string":"Fudan University , Shanghai 200433 , China","institution_ids":["https://openalex.org/I24943067"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"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":0.8970000147819519,"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":0.8970000147819519,"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/T11166","display_name":"Analytic Number Theory Research","score":0.05550000071525574,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/T11151","display_name":"Advanced Topology and Set Theory","score":0.005499999970197678,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/counterexample","display_name":"counterexample","score":0.8019999861717224},{"id":"https://openalex.org/keywords/diophantine-approximation","display_name":"Diophantine approximation","score":0.7350000143051147},{"id":"https://openalex.org/keywords/hausdorff-dimension","display_name":"Hausdorff dimension","score":0.9319999814033508},{"id":"https://openalex.org/keywords/regular-curves","display_name":"regular curves","score":0.5210000276565552}],"concepts":[{"id":"https://openalex.org/C162838799","wikidata":"https://www.wikidata.org/wiki/Q596077","display_name":"Counterexample","level":2,"score":0.8733999729156494},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8730000257492065},{"id":"https://openalex.org/C68693459","wikidata":"https://www.wikidata.org/wiki/Q657586","display_name":"Hyperplane","level":2,"score":0.8169999718666077},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.7408000230789185},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6482999920845032},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5063999891281128},{"id":"https://openalex.org/C194198291","wikidata":"https://www.wikidata.org/wiki/Q565186","display_name":"Hausdorff dimension","level":2,"score":0.49540001153945923},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.4438999891281128},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.4375},{"id":"https://openalex.org/C191399826","wikidata":"https://www.wikidata.org/wiki/Q326908","display_name":"Hausdorff space","level":2,"score":0.3765000104904175},{"id":"https://openalex.org/C189950617","wikidata":"https://www.wikidata.org/wiki/Q937228","display_name":"Property (philosophy)","level":2,"score":0.3601999878883362},{"id":"https://openalex.org/C2777026412","wikidata":"https://www.wikidata.org/wiki/Q2684591","display_name":"Statement (logic)","level":2,"score":0.3474999964237213},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.33809998631477356},{"id":"https://openalex.org/C2777487392","wikidata":"https://www.wikidata.org/wiki/Q481201","display_name":"Absolute (philosophy)","level":2,"score":0.33500000834465027},{"id":"https://openalex.org/C17327563","wikidata":"https://www.wikidata.org/wiki/Q1991405","display_name":"Nowhere dense set","level":3,"score":0.3215000033378601},{"id":"https://openalex.org/C94020503","wikidata":"https://www.wikidata.org/wiki/Q12916","display_name":"Real number","level":2,"score":0.27900001406669617},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.2524999976158142},{"id":"https://openalex.org/C141898687","wikidata":"https://www.wikidata.org/wiki/Q1501997","display_name":"Hausdorff distance","level":2,"score":0.25}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2608.24401","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.24401","pdf_url":"https://arxiv.org/pdf/2608.24401","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.2608.24401","doi":"https://doi.org/10.48550/arxiv.2608.24401","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.24401","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:2608.24401","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.24401","pdf_url":"https://arxiv.org/pdf/2608.24401","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/W7208764988.pdf","grobid_xml":"https://content.openalex.org/works/W7208764988.grobid-xml"},"referenced_works_count":4,"referenced_works":["https://openalex.org/W4409738244","https://openalex.org/W4414763208","https://openalex.org/W7136519464","https://openalex.org/W7162569440"],"related_works":["https://openalex.org/W7204494580","https://openalex.org/W7136519464","https://openalex.org/W4288725935","https://openalex.org/W2066922393","https://openalex.org/W3209437437","https://openalex.org/W7095700868","https://openalex.org/W2099449228","https://openalex.org/W2127285465","https://openalex.org/W7097163987","https://openalex.org/W1878171348"],"abstract_inverted_index":{"In":[0,79,90],"this":[1],"paper,":[2],"we":[3,92],"prove":[4,93],"that":[5,18,45,94],"the":[6,11,20,50,81,118],"set":[7,21,51,107],"of":[8,22,24,52,108,110],"counterexamples":[9],"to":[10],"uniform":[12],"Littlewood's":[13],"conjecture":[14],"proposed":[15],"in":[16,88],"Bandi-Fregoli-Kleinbock,":[17],"is,":[19],"pairs":[23,54],"real":[25,53],"numbers":[26],"$(x,y)$":[27,55],"satisfying":[28,56],"$$":[29,38,57,66],"\\limsup_{Q\\to":[30],"+\\infty}\\":[31,59],"Q\\cdot":[32],"\\min_{1\\leq":[33,61],"q\\leq":[34,62],"Q}\\langle":[35],"qx\\rangle\\langle":[36,64],"qy\\rangle>0,":[37,65],"is":[39,67,113],"hyperplane":[40,68],"absolute":[41,69,98,115],"winning.":[42],"We":[43,120],"show":[44],"a":[46],"stronger":[47],"statement":[48],"holds:":[49],"\\liminf_{m\\to":[58],"Q_m\\cdot":[60],"Q_m}\\langle":[63],"winning":[70,99,116],"if":[71],"$Q_{m+1}":[72],"\\gg":[73],"Q_m^\u03c4$":[74],"for":[75,125],"some":[76],"$\u03c4>":[77],"1$.":[78],"particular,":[80],"above":[82],"sets":[83,96],"have":[84],"full":[85],"Hausdorff":[86],"dimension":[87],"$\\mathbb{R}^2$.":[89],"addition,":[91],"these":[95],"are":[97],"on":[100,117],"every":[101],"regular":[102],"$C^2$":[103],"planar":[104],"curve":[105],"whose":[106],"points":[109],"nonzero":[111],"curvature":[112],"itself":[114],"curve.":[119],"also":[121],"establish":[122],"analogous":[123],"results":[124],"certain":[126],"lines.":[127]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-05T00:00:00"}
