{"id":"https://openalex.org/W7204475771","doi":"https://doi.org/10.48550/arxiv.2608.25349","title":"Well and badly approximable sets, and rapid winning","display_name":"Well and badly approximable sets, and rapid winning","publication_year":2026,"publication_date":"2026-08-26","ids":{"openalex":"https://openalex.org/W7204475771","doi":"https://doi.org/10.48550/arxiv.2608.25349"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2608.25349","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.25349","pdf_url":"https://arxiv.org/pdf/2608.25349","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/2608.25349","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5064739044","display_name":"Mumtaz Hussain","orcid":"https://orcid.org/0000-0001-5621-9341","observed_orcids":["https://orcid.org/0000-0001-5621-9341"]},"institutions":[{"id":"https://openalex.org/I196829312","display_name":"La Trobe University","ror":"https://ror.org/01rxfrp27","country_code":"AU","type":"education","lineage":["https://openalex.org/I196829312"]}],"countries":["AU"],"is_corresponding":false,"raw_author_name":"Hussain, Mumtaz","raw_affiliation_strings":["434 Hanover Ln , Irving , TX 75062 , USA","Department of Mathematical and Physical Sciences , La Trobe University , Bendigo 3552 , Australia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"434 Hanover Ln , Irving , TX 75062 , USA","institution_ids":[]},{"raw_affiliation_string":"Department of Mathematical and Physical Sciences , La Trobe University , Bendigo 3552 , Australia","institution_ids":["https://openalex.org/I196829312"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5148437288","display_name":"David Simmons","orcid":"https://orcid.org/0000-0002-9136-6635","observed_orcids":["https://orcid.org/0000-0002-9136-6635"]},"institutions":[{"id":"https://openalex.org/I196829312","display_name":"La Trobe University","ror":"https://ror.org/01rxfrp27","country_code":"AU","type":"education","lineage":["https://openalex.org/I196829312"]}],"countries":["AU"],"is_corresponding":false,"raw_author_name":"Simmons, David","raw_affiliation_strings":["Department of Mathematical and Physical Sciences , La Trobe University , Bendigo 3552 , Australia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematical and Physical Sciences , La Trobe University , Bendigo 3552 , Australia","institution_ids":["https://openalex.org/I196829312"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I196829312"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":3,"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":0.8920999765396118,"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":0.8920999765396118,"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":0.06319999694824219,"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":0.008100000210106373,"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/badly-approximable-numbers","display_name":"badly approximable numbers","score":0.9769999980926514},{"id":"https://openalex.org/keywords/inhomogeneous-diophantine-approximation","display_name":"inhomogeneous Diophantine approximation","score":0.9020000100135803},{"id":"https://openalex.org/keywords/hausdorff-dimension","display_name":"Hausdorff dimension","score":0.9760000109672546},{"id":"https://openalex.org/keywords/approximation-functions","display_name":"approximation functions","score":0.6480000019073486},{"id":"https://openalex.org/keywords/diophantine-approximation","display_name":"Diophantine approximation","score":0.7200000286102295}],"concepts":[{"id":"https://openalex.org/C194198291","wikidata":"https://www.wikidata.org/wiki/Q565186","display_name":"Hausdorff dimension","level":2,"score":0.8514999747276306},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.755299985408783},{"id":"https://openalex.org/C64543145","wikidata":"https://www.wikidata.org/wiki/Q162942","display_name":"Intersection (aeronautics)","level":2,"score":0.7049999833106995},{"id":"https://openalex.org/C191399826","wikidata":"https://www.wikidata.org/wiki/Q326908","display_name":"Hausdorff space","level":2,"score":0.6155999898910522},{"id":"https://openalex.org/C115311070","wikidata":"https://www.wikidata.org/wiki/Q5347255","display_name":"Effective dimension","level":3,"score":0.5934000015258789},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.5270000100135803},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.505299985408783},{"id":"https://openalex.org/C141898687","wikidata":"https://www.wikidata.org/wiki/Q1501997","display_name":"Hausdorff distance","level":2,"score":0.5051000118255615},{"id":"https://openalex.org/C189950617","wikidata":"https://www.wikidata.org/wiki/Q937228","display_name":"Property (philosophy)","level":2,"score":0.487199991941452},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.47929999232292175},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.4657999873161316},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4546999931335449},{"id":"https://openalex.org/C28235433","wikidata":"https://www.wikidata.org/wiki/Q5277259","display_name":"Dimension function","level":3,"score":0.42719998955726624},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.4221000075340271},{"id":"https://openalex.org/C143924541","wikidata":"https://www.wikidata.org/wiki/Q258374","display_name":"Outer measure","level":5,"score":0.3822000026702881},{"id":"https://openalex.org/C73225184","wikidata":"https://www.wikidata.org/wiki/Q1591095","display_name":"Hausdorff measure","level":3,"score":0.373199999332428},{"id":"https://openalex.org/C80661125","wikidata":"https://www.wikidata.org/wiki/Q5404299","display_name":"Packing dimension","level":5,"score":0.353300005197525},{"id":"https://openalex.org/C102966492","wikidata":"https://www.wikidata.org/wiki/Q1227061","display_name":"Diophantine approximation","level":3,"score":0.31700000166893005},{"id":"https://openalex.org/C184264201","wikidata":"https://www.wikidata.org/wiki/Q21199","display_name":"Natural number","level":2,"score":0.31189998984336853},{"id":"https://openalex.org/C84496893","wikidata":"https://www.wikidata.org/wiki/Q1135782","display_name":"Urysohn and completely Hausdorff spaces","level":4,"score":0.2867000102996826},{"id":"https://openalex.org/C2776608160","wikidata":"https://www.wikidata.org/wiki/Q4785462","display_name":"Natural (archaeology)","level":2,"score":0.2761000096797943}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2608.25349","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.25349","pdf_url":"https://arxiv.org/pdf/2608.25349","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.2608.25349","doi":"https://doi.org/10.48550/arxiv.2608.25349","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.25349","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:2608.25349","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.25349","pdf_url":"https://arxiv.org/pdf/2608.25349","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":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7204475771.pdf","grobid_xml":"https://content.openalex.org/works/W7204475771.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W4405034924","https://openalex.org/W7160596560","https://openalex.org/W4416056109","https://openalex.org/W1995208920","https://openalex.org/W2146130936","https://openalex.org/W2148911164","https://openalex.org/W4407807006","https://openalex.org/W7127337376","https://openalex.org/W4289731193","https://openalex.org/W272596718"],"abstract_inverted_index":{"The":[0],"set":[1,14],"of":[2,15,30,41,45,105],"$\u03c4$-approximable":[3],"numbers,":[4,19],"$\\mathcal":[5,59],"W(\u03c4)$,":[6],"has":[7,21],"genuinely":[8],"fractional":[9],"Hausdorff":[10,23,28,82],"dimension,":[11,83,117],"whereas":[12],"the":[13,27,35,42,84,96,103,114],"inhomogeneously":[16],"badly":[17],"approximable":[18],"$\\Bad^\u03b3$,":[20],"full":[22,81],"dimension.":[24],"We":[25],"determine":[26],"dimension":[29,98],"their":[31],"intersection":[32],"by":[33,71],"introducing":[34],"$\u03a8$-rapid":[36,63,85],"game,":[37],"a":[38,66,90],"scale-sensitive":[39],"refinement":[40],"rapid":[43],"game":[44,86],"Hatefi":[46],"and":[47],"Simmons":[48],"(preprint":[49],"2024).":[50],"For":[51],"every":[52],"approximation":[53],"function":[54],"$\u03c8$,":[55],"we":[56,112],"prove":[57],"that":[58,95,118],"W(\u03c8)\\cap\\Bad^\u03b3$":[60],"is":[61,87],"strong":[62],"winning":[64,77],"for":[65,109],"natural":[67],"gauge":[68],"$\u03a8$":[69],"determined":[70],"$\u03c8$.":[72],"Unlike":[73],"Schmidt-type":[74],"games,":[75],"whose":[76],"property":[78],"always":[79],"implies":[80],"calibrated":[88],"to":[89],"prescribed":[91],"Diophantine":[92],"scale,":[93],"so":[94],"resulting":[97],"bound":[99],"depends":[100],"explicitly":[101],"on":[102],"decay":[104],"$\u03a8$.":[106],"In":[107],"particular,":[108],"$\u03c8(q)=q^{-\u03c4},":[110],"\u03c4\\ge1,$":[111],"recover":[113],"exact":[115],"Jarn\u00edk--Besicovitch":[116],"is,":[119],"$$":[120],"\\HD\\bigl(\\mathcal":[121],"W(\u03c4)\\cap\\Bad^\u03b3\\bigr)=\\frac{2}{\u03c4+1}.$$":[122]},"counts_by_year":[{"year":2026,"cited_by_count":3}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-08-28T00:00:00"}
