{"id":"https://openalex.org/W1526069328","doi":"https://doi.org/10.37236/1933","title":"Indecomposable Tilings of the Integers with Exponentially Long Periods","display_name":"Indecomposable Tilings of the Integers with Exponentially Long Periods","publication_year":2005,"publication_date":"2005-07-29","ids":{"openalex":"https://openalex.org/W1526069328","doi":"https://doi.org/10.37236/1933","mag":"1526069328"},"language":"en","primary_location":{"id":"doi:10.37236/1933","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1933","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v12i1r36/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v12i1r36/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5068837614","display_name":"John Steinberger","orcid":null},"institutions":[{"id":"https://openalex.org/I84218800","display_name":"University of California, Davis","ror":"https://ror.org/05rrcem69","country_code":"US","type":"education","lineage":["https://openalex.org/I84218800"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"John P. Steinberger","raw_affiliation_strings":["Department of Mathematics UC Davis, California, USA"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics UC Davis, California, USA","institution_ids":["https://openalex.org/I84218800"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5068837614"],"corresponding_institution_ids":["https://openalex.org/I84218800"],"apc_list":null,"apc_paid":null,"fwci":0.8113,"has_fulltext":true,"cited_by_count":9,"citation_normalized_percentile":{"value":0.68842672,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":95},"biblio":{"volume":"12","issue":"1","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11887","display_name":"Quasicrystal Structures and Properties","score":0.9979000091552734,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11887","display_name":"Quasicrystal Structures and Properties","score":0.9979000091552734,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11797","display_name":"graph theory and CDMA systems","score":0.9969000220298767,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11567","display_name":"semigroups and automata theory","score":0.9811000227928162,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/multiset","display_name":"Multiset","score":0.9391615390777588},{"id":"https://openalex.org/keywords/indecomposable-module","display_name":"Indecomposable module","score":0.8878790140151978},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.8422433137893677},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8317881226539612},{"id":"https://openalex.org/keywords/disjoint-sets","display_name":"Disjoint sets","score":0.7965025901794434},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.6184350252151489},{"id":"https://openalex.org/keywords/infinity","display_name":"Infinity","score":0.43639475107192993},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.43368983268737793}],"concepts":[{"id":"https://openalex.org/C2779623528","wikidata":"https://www.wikidata.org/wiki/Q864377","display_name":"Multiset","level":2,"score":0.9391615390777588},{"id":"https://openalex.org/C157480366","wikidata":"https://www.wikidata.org/wiki/Q2499534","display_name":"Indecomposable module","level":2,"score":0.8878790140151978},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8422433137893677},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8317881226539612},{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.7965025901794434},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.6184350252151489},{"id":"https://openalex.org/C7321624","wikidata":"https://www.wikidata.org/wiki/Q205","display_name":"Infinity","level":2,"score":0.43639475107192993},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.43368983268737793},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.37236/1933","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1933","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v12i1r36/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.135.383","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.135.383","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.math.ucdavis.edu/~jpsteinb/period.ps","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.88.5792","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.88.5792","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.emis.de/journals/EJC/Volume_12/PDF/v12i1r36.pdf","raw_type":"text"}],"best_oa_location":{"id":"doi:10.37236/1933","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1933","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v12i1r36/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G837910662","display_name":"Vertical Integration of Research and Education in the Mathematical Sciences - VIGRE: Research Focus Groups in Mathematics","funder_award_id":"0135345","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1526069328.pdf","grobid_xml":"https://content.openalex.org/works/W1526069328.grobid-xml"},"referenced_works_count":9,"referenced_works":["https://openalex.org/W2016858362","https://openalex.org/W2040410687","https://openalex.org/W2083844723","https://openalex.org/W2100678102","https://openalex.org/W2108918469","https://openalex.org/W2128543548","https://openalex.org/W2131367862","https://openalex.org/W2145609240","https://openalex.org/W6602577056"],"related_works":["https://openalex.org/W4300455649","https://openalex.org/W2296588302","https://openalex.org/W4328119206","https://openalex.org/W2033245919","https://openalex.org/W4252409532","https://openalex.org/W2006067547","https://openalex.org/W9228044","https://openalex.org/W3143753676","https://openalex.org/W2017752594","https://openalex.org/W2072510267"],"abstract_inverted_index":{"Let":[0],"$A$":[1,40],"be":[2,17,27,55],"a":[3,88],"finite":[4],"multiset":[5,10],"of":[6,11,20,36,39,42,45,61,90,102,133],"integers.":[7],"A":[8],"second":[9],"integers":[12],"$T$":[13,49],"is":[14,50,129],"said":[15],"to":[16,75,122],"an":[18,37,43,98],"$A$-tiling":[19],"level":[21],"$d$":[22,31],"if":[23,52],"every":[24],"integer":[25],"can":[26],"expressed":[28],"in":[29],"exactly":[30],"ways":[32],"as":[33,57],"the":[34,58],"sum":[35],"element":[38,44],"and":[41,124],"$T$.":[46],"The":[47],"set":[48],"indecomposable":[51,77,99],"it":[53],"cannot":[54],"written":[56],"disjoint":[59],"union":[60],"two":[62],"proper":[63],"subsets":[64],"that":[65,79,94],"are":[66],"also":[67],"$A$-tilings.":[68],"In":[69],"this":[70],"paper":[71],"we":[72,86],"show":[73],"how":[74],"construct":[76],"tilings":[78],"have":[80],"exponentially":[81],"long":[82],"periods.":[83],"More":[84],"precisely,":[85],"give":[87],"sequence":[89],"multisets":[91],"$(A_k)_{k=1}^{\\infty}$":[92],"such":[93],"each":[95],"$A_k$":[96],"admits":[97],"tiling":[100],"$T_k$":[101],"period":[103],"greater":[104],"than":[105],"$e^{c\\root":[106],"3\\of{n_k\\log(n_k)}}$":[107],"where":[108,125],"$n_k":[109],"=":[110,113],"{\\rm":[111],"diam}(A_k)":[112],"\\max\\{j":[114],"\\in":[115,119],"A_k\\}":[116],"-":[117],"\\min\\{j":[118],"A_k\\}$":[120],"tends":[121],"infinity":[123],"$c":[126],">":[127],"0$":[128],"some":[130],"constant":[131],"independent":[132],"$k$.":[134]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}