{"id":"https://openalex.org/W7214048563","doi":"https://doi.org/10.48550/arxiv.2609.24412","title":"Asymptotic Behavior of Iterated Sets of Remainders","display_name":"Asymptotic Behavior of Iterated Sets of Remainders","publication_year":2026,"publication_date":"2026-09-21","ids":{"openalex":"https://openalex.org/W7214048563","doi":"https://doi.org/10.48550/arxiv.2609.24412"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.24412","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.24412","pdf_url":"https://arxiv.org/pdf/2609.24412","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/2609.24412","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5141295258","display_name":"Omkar Baraskar","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I151746483","display_name":"University of Waterloo","ror":"https://ror.org/01aff2v68","country_code":"CA","type":"education","lineage":["https://openalex.org/I151746483"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Baraskar, Omkar","raw_affiliation_strings":["Cheriton School of Computer Science University of Waterloo ON N2L 3G1 , Canada"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Cheriton School of Computer Science University of Waterloo ON N2L 3G1 , Canada","institution_ids":["https://openalex.org/I151746483"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5066946498","display_name":"Prashant Gokhale","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I135310074","display_name":"University of Wisconsin\u2013Madison","ror":"https://ror.org/01y2jtd41","country_code":"US","type":"education","lineage":["https://openalex.org/I135310074"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Gokhale, Prashant","raw_affiliation_strings":["University of Wisconsin-Madison"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Wisconsin-Madison","institution_ids":["https://openalex.org/I135310074"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5110990128","display_name":"Sarvagya Jain","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I155660961","display_name":"University of Turku","ror":"https://ror.org/05vghhr25","country_code":"FI","type":"education","lineage":["https://openalex.org/I155660961"]}],"countries":["FI"],"is_corresponding":false,"raw_author_name":"Jain, Sarvagya","raw_affiliation_strings":["Department of Mathematics and Statistics University of Turku 20014 Turku , Finland"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics University of Turku 20014 Turku , Finland","institution_ids":["https://openalex.org/I155660961"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5019170764","display_name":"Adam Kie\u017cun","orcid":"https://orcid.org/0000-0002-6634-1073","observed_orcids":["https://orcid.org/0000-0002-6634-1073"]},"institutions":[{"id":"https://openalex.org/I4210111288","display_name":"Meta (United Kingdom)","ror":"https://ror.org/020ye1821","country_code":"GB","type":"company","lineage":["https://openalex.org/I4210111288","https://openalex.org/I4210114444"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Kie\u017cun, Adam","raw_affiliation_strings":["Meta Superintelligence Labs"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Meta Superintelligence Labs","institution_ids":["https://openalex.org/I4210111288"]}]}],"institutions":[],"countries_distinct_count":4,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":2,"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/T11166","display_name":"Analytic Number Theory Research","score":0.9883000254631042,"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"}},"topics":[{"id":"https://openalex.org/T11166","display_name":"Analytic Number Theory Research","score":0.9883000254631042,"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/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.0017999999690800905,"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/T14500","display_name":"Smarandache Functions and Recreational Number Theory","score":0.0017000000225380063,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/asymptotic-behavior","display_name":"asymptotic behavior","score":0.8339999914169312},{"id":"https://openalex.org/keywords/modular-arithmetic","display_name":"modular arithmetic","score":0.34599998593330383}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8414000272750854},{"id":"https://openalex.org/C39613435","wikidata":"https://www.wikidata.org/wiki/Q846677","display_name":"Remainder","level":2,"score":0.7613999843597412},{"id":"https://openalex.org/C140479938","wikidata":"https://www.wikidata.org/wiki/Q5254619","display_name":"Iterated function","level":2,"score":0.7210000157356262},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.6854000091552734},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.6686000227928162},{"id":"https://openalex.org/C143724316","wikidata":"https://www.wikidata.org/wiki/Q312468","display_name":"Series (stratigraphy)","level":2,"score":0.569599986076355},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.5473999977111816},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.5401999950408936},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.47130000591278076},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4706999957561493},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.3675999939441681},{"id":"https://openalex.org/C11683690","wikidata":"https://www.wikidata.org/wiki/Q752726","display_name":"Asymptotic expansion","level":2,"score":0.3666999936103821},{"id":"https://openalex.org/C46875033","wikidata":"https://www.wikidata.org/wiki/Q1244890","display_name":"Rational number","level":2,"score":0.3312999904155731},{"id":"https://openalex.org/C136979486","wikidata":"https://www.wikidata.org/wiki/Q773483","display_name":"Existential quantification","level":2,"score":0.3068000078201294},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3018999993801117},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.2921999990940094},{"id":"https://openalex.org/C2778753846","wikidata":"https://www.wikidata.org/wiki/Q6554239","display_name":"Link (geometry)","level":2,"score":0.2793999910354614},{"id":"https://openalex.org/C184264201","wikidata":"https://www.wikidata.org/wiki/Q21199","display_name":"Natural number","level":2,"score":0.26919999718666077},{"id":"https://openalex.org/C61445026","wikidata":"https://www.wikidata.org/wiki/Q217608","display_name":"Fixed point","level":2,"score":0.263700008392334}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.24412","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.24412","pdf_url":"https://arxiv.org/pdf/2609.24412","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.2609.24412","doi":"https://doi.org/10.48550/arxiv.2609.24412","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.24412","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:2609.24412","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.24412","pdf_url":"https://arxiv.org/pdf/2609.24412","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/W7214048563.pdf","grobid_xml":"https://content.openalex.org/works/W7214048563.grobid-xml"},"referenced_works_count":1,"referenced_works":["https://openalex.org/W4309217119"],"related_works":["https://openalex.org/W4414450151","https://openalex.org/W2313589666","https://openalex.org/W4309217119","https://openalex.org/W7223044198","https://openalex.org/W4409708291","https://openalex.org/W4322745459","https://openalex.org/W7213939828","https://openalex.org/W2012123466","https://openalex.org/W7220704611","https://openalex.org/W2735135343"],"abstract_inverted_index":{"For":[0],"a":[1,34,69,92],"positive":[2],"integer":[3,120],"$n$,":[4],"let":[5],"$$S_0(n)=\\{1,2,\\ldots,\\lfloor":[6],"n/2\\rfloor\\},\\qquad":[7],"S_{j+1}(n)=\\{n\\bmod":[8],"k:k\\in":[9],"S_j(n)\\setminus\\{0\\}\\},$$":[10],"and":[11,40,113],"put":[12],"$s_j(n)":[13],":=":[14,103],"|S_j(n)|$.":[15],"The":[16],"sets":[17,75],"defined":[18],"above":[19],"arise":[20],"naturally":[21,78],"in":[22,63],"the":[23,26,29,49,64,81,84,87,98],"study":[24,82,97],"of":[25,28,33,72,83,86,91,101],"length":[27,85],"Pierce":[30],"series":[31,89],"expansion":[32,90],"rational":[35,93],"number.":[36,94],"In":[37,56],"\\cite{Ba-Vu},":[38],"Baraskar":[39],"Vukusic":[41],"conjectured":[42],"that":[43,77,107],"for":[44,111,117],"every":[45,118],"fixed":[46,119],"$j\\geq":[47],"2$,":[48],"limit":[50],"$$\\lim_{n\\to\\infty}":[51],"s_j(n)/n$$":[52],"does":[53],"not":[54],"exist.":[55],"this":[57,61],"paper,":[58],"we":[59,67],"prove":[60],"conjecture":[62],"affirmative.":[65],"Moreover,":[66],"define":[68],"new":[70],"class":[71],"iterated":[73],"remainder":[74],"$T_j(n)$":[76],"arises":[79],"from":[80],"Engel":[88],"We":[95,105],"analogously":[96],"asymptotic":[99],"behavior":[100],"$t_j(n)":[102],"|T_j(n)|$.":[104],"show":[106],"$$\\lim_{n\\to\\infty}\\frac{t_j(n)}n$$":[108],"exists":[109],"precisely":[110],"$j\\in\\{0,1\\}$":[112],"fails":[114],"to":[115],"exist":[116],"$j\\geq2$.":[121]},"counts_by_year":[{"year":2026,"cited_by_count":2}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-23T00:00:00"}
