{"id":"https://openalex.org/W7125673617","doi":"https://doi.org/10.48550/arxiv.2601.16571","title":"The inverse of the star discrepancy of a union of randomly shifted Korobov rank-1 lattice point sets depends polynomially on the dimension","display_name":"The inverse of the star discrepancy of a union of randomly shifted Korobov rank-1 lattice point sets depends polynomially on the dimension","publication_year":2026,"publication_date":"2026-01-23","ids":{"openalex":"https://openalex.org/W7125673617","doi":"https://doi.org/10.48550/arxiv.2601.16571"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2601.16571","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2601.16571","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,"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"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2601.16571","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5102627191","display_name":"Jiarui Du","orcid":"https://orcid.org/0009-0008-1956-6537"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Du, Jiarui","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5123846348","display_name":"Josef Dick","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Dick, Josef","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":false,"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/T12404","display_name":"Mathematical Approximation and Integration","score":0.9697999954223633,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"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/T12404","display_name":"Mathematical Approximation and Integration","score":0.9697999954223633,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"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/T11435","display_name":"Polynomial and algebraic computation","score":0.00559999980032444,"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"}},{"id":"https://openalex.org/T12056","display_name":"Markov Chains and Monte Carlo Methods","score":0.002400000113993883,"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/lattice","display_name":"Lattice (music)","score":0.486299991607666},{"id":"https://openalex.org/keywords/star","display_name":"Star (game theory)","score":0.4763999879360199},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.4636000096797943},{"id":"https://openalex.org/keywords/integer-lattice","display_name":"Integer lattice","score":0.4375},{"id":"https://openalex.org/keywords/multiset","display_name":"Multiset","score":0.42829999327659607},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.42800000309944153},{"id":"https://openalex.org/keywords/inverse","display_name":"Inverse","score":0.41370001435279846},{"id":"https://openalex.org/keywords/hausdorff-dimension","display_name":"Hausdorff dimension","score":0.40470001101493835},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.38940000534057617}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8440999984741211},{"id":"https://openalex.org/C2781204021","wikidata":"https://www.wikidata.org/wiki/Q6497091","display_name":"Lattice (music)","level":2,"score":0.486299991607666},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.48069998621940613},{"id":"https://openalex.org/C2780897414","wikidata":"https://www.wikidata.org/wiki/Q7600592","display_name":"Star (game theory)","level":2,"score":0.4763999879360199},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.4636000096797943},{"id":"https://openalex.org/C142885752","wikidata":"https://www.wikidata.org/wiki/Q1252145","display_name":"Integer lattice","level":3,"score":0.4375},{"id":"https://openalex.org/C2779623528","wikidata":"https://www.wikidata.org/wiki/Q864377","display_name":"Multiset","level":2,"score":0.42829999327659607},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.42800000309944153},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.41370001435279846},{"id":"https://openalex.org/C194198291","wikidata":"https://www.wikidata.org/wiki/Q565186","display_name":"Hausdorff dimension","level":2,"score":0.40470001101493835},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.38940000534057617},{"id":"https://openalex.org/C50820777","wikidata":"https://www.wikidata.org/wiki/Q213723","display_name":"Hypercube","level":2,"score":0.38749998807907104},{"id":"https://openalex.org/C54732982","wikidata":"https://www.wikidata.org/wiki/Q1415345","display_name":"Modulo","level":2,"score":0.3790000081062317},{"id":"https://openalex.org/C165464430","wikidata":"https://www.wikidata.org/wiki/Q1570441","display_name":"Parameterized complexity","level":2,"score":0.3781999945640564},{"id":"https://openalex.org/C115311070","wikidata":"https://www.wikidata.org/wiki/Q5347255","display_name":"Effective dimension","level":3,"score":0.37610000371932983},{"id":"https://openalex.org/C169654258","wikidata":"https://www.wikidata.org/wiki/Q12479","display_name":"Number theory","level":2,"score":0.35580000281333923},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.3513999879360199},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3506999909877777},{"id":"https://openalex.org/C88871306","wikidata":"https://www.wikidata.org/wiki/Q7208287","display_name":"Point process","level":2,"score":0.33559998869895935},{"id":"https://openalex.org/C102966492","wikidata":"https://www.wikidata.org/wiki/Q1227061","display_name":"Diophantine approximation","level":3,"score":0.2635999917984009},{"id":"https://openalex.org/C41608201","wikidata":"https://www.wikidata.org/wiki/Q980509","display_name":"Embedding","level":2,"score":0.25589999556541443},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.25540000200271606},{"id":"https://openalex.org/C207864730","wikidata":"https://www.wikidata.org/wiki/Q179467","display_name":"Fourier series","level":2,"score":0.25200000405311584},{"id":"https://openalex.org/C195956108","wikidata":"https://www.wikidata.org/wiki/Q7268362","display_name":"Quadratic growth","level":2,"score":0.25110000371932983}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2601.16571","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2601.16571","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,"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":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2601.16571","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2601.16571","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,"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"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"The":[0],"inverse":[1,177],"of":[2,13,21,59,84,90,105,156,171,178,215],"the":[3,10,70,82,88,148,168,176,179,189,193,201],"star":[4,19,149,180],"discrepancy,":[5],"$N(\u03b5,":[6],"s)$,":[7],"defined":[8],"as":[9,102],"minimum":[11],"number":[12,170],"points":[14],"required":[15],"to":[16,30,211],"achieve":[17,69],"a":[18,56,112,117,154,209,212],"discrepancy":[20,150,181],"at":[22],"most":[23],"$\u03b5$":[24],"in":[25,87,144],"dimension":[26,72,190],"$s$,":[27],"is":[28,151,167],"known":[29],"depend":[31],"linearly":[32],"on":[33,188],"$s$.":[34,191],"However,":[35],"explicit":[36],"constructions":[37],"achieving":[38],"this":[39,78,85],"optimal":[40,71,205],"linear":[41],"dependence":[42,73],"remain":[43],"elusive.":[44],"Recently,":[45],"Dick":[46],"and":[47,221],"Pillichshammer":[48],"(2025)":[49],"made":[50],"significant":[51],"progress":[52],"by":[53,153,218],"showing":[54],"that":[55,143,175],"multiset":[57,103],"union":[58],"randomly":[60],"digitally":[61],"shifted":[62],"Korobov":[63,106],"polynomial":[64],"lattice":[65,108],"point":[66,99,109,206],"sets":[67,100,110,185,207],"almost":[68],"with":[74,132,162],"high":[75,163],"probability.":[76],"In":[77],"paper,":[79],"we":[80],"investigate":[81],"analog":[83],"result":[86],"setting":[89],"classical":[91],"integer":[92,130,219],"arithmetic":[93],"using":[94],"Fourier":[95],"analysis.":[96],"We":[97,115,141],"analyze":[98],"constructed":[101],"unions":[104],"rank-1":[107],"modulo":[111],"prime":[113],"$N$.":[114],"provide":[116],"comprehensive":[118],"analysis":[119],"covering":[120],"four":[121,146],"distinct":[122],"construction":[123],"scenarios,":[124],"combining":[125],"either":[126,133],"random":[127,222],"or":[128,137],"fixed":[129],"generators":[131,220],"continuous":[134],"torus":[135],"shifts":[136],"discrete":[138],"grid":[139],"shifts.":[140,223],"prove":[142],"all":[145],"cases,":[147],"bounded":[152],"term":[155],"order":[157],"$O(s":[158],"\\log(N_{tot})":[159],"/":[160],"\\sqrt{N_{tot}})$":[161],"probability,":[164],"where":[165],"$N_{tot}$":[166],"total":[169],"points.":[172],"This":[173],"implies":[174],"for":[182,204],"these":[183],"structured":[184],"depends":[186],"quadratically":[187],"While":[192],"proofs":[194],"are":[195],"probabilistic,":[196],"our":[197],"results":[198],"significantly":[199],"reduce":[200],"search":[202],"space":[203],"from":[208],"continuum":[210],"finite":[213],"set":[214],"candidates":[216],"parameterized":[217]},"counts_by_year":[],"updated_date":"2026-07-28T07:46:37.118299","created_date":"2026-01-27T00:00:00"}
