{"id":"https://openalex.org/W3000339888","doi":"https://doi.org/10.4230/lipics.cpm.2026.6","title":"A proof that a word of length n has less than 1.5n distinct squares","display_name":"A proof that a word of length n has less than 1.5n distinct squares","publication_year":2020,"publication_date":"2020-01-07","ids":{"openalex":"https://openalex.org/W3000339888","doi":"https://doi.org/10.4230/lipics.cpm.2026.6","mag":"3000339888"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2001.02996","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2001.02996","pdf_url":"https://arxiv.org/pdf/2001.02996","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"text"},"type":"preprint","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2001.02996","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5005877692","display_name":"Adrien Thierry","orcid":"https://orcid.org/0000-0002-5765-3585"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Thierry, Adrien","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5005877692"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.2914,"has_fulltext":true,"cited_by_count":4,"citation_normalized_percentile":{"value":0.6048371,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":96},"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/T11567","display_name":"semigroups and automata theory","score":0.9991999864578247,"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"}},"topics":[{"id":"https://openalex.org/T11567","display_name":"semigroups and automata theory","score":0.9991999864578247,"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/T11130","display_name":"Coding theory and cryptography","score":0.9986000061035156,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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/T11269","display_name":"Algorithms and Data Compression","score":0.9969000220298767,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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/combinatorics","display_name":"Combinatorics","score":0.7945430278778076},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7683272361755371},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.7618708610534668},{"id":"https://openalex.org/keywords/lemma","display_name":"Lemma (botany)","score":0.7597495913505554},{"id":"https://openalex.org/keywords/word","display_name":"Word (group theory)","score":0.7459510564804077},{"id":"https://openalex.org/keywords/word-length","display_name":"Word length","score":0.5878733396530151},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4376169443130493},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.12468922138214111},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.07587683200836182},{"id":"https://openalex.org/keywords/natural-language-processing","display_name":"Natural language processing","score":0.0577491819858551},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.04963332414627075}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7945430278778076},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7683272361755371},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.7618708610534668},{"id":"https://openalex.org/C2777759810","wikidata":"https://www.wikidata.org/wiki/Q149316","display_name":"Lemma (botany)","level":3,"score":0.7597495913505554},{"id":"https://openalex.org/C90805587","wikidata":"https://www.wikidata.org/wiki/Q10944557","display_name":"Word (group theory)","level":2,"score":0.7459510564804077},{"id":"https://openalex.org/C3019641298","wikidata":"https://www.wikidata.org/wiki/Q625642","display_name":"Word length","level":2,"score":0.5878733396530151},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4376169443130493},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.12468922138214111},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.07587683200836182},{"id":"https://openalex.org/C204321447","wikidata":"https://www.wikidata.org/wiki/Q30642","display_name":"Natural language processing","level":1,"score":0.0577491819858551},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.04963332414627075},{"id":"https://openalex.org/C46757340","wikidata":"https://www.wikidata.org/wiki/Q43238","display_name":"Poaceae","level":2,"score":0.0},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"pmh:oai:arXiv.org:2001.02996","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2001.02996","pdf_url":"https://arxiv.org/pdf/2001.02996","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"text"},{"id":"pmh:oai:drops-oai.dagstuhl.de:25932","is_oa":true,"landing_page_url":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2026.6","pdf_url":"https://drops.dagstuhl.de/storage/00lipics/lipics-vol369-cpm2026/LIPIcs.CPM.2026.6/LIPIcs.CPM.2026.6.pdf","source":{"id":"https://openalex.org/S4377196569","display_name":"DROPS (Schloss Dagstuhl \u2013 Leibniz Center for Informatics)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I2799853480","host_organization_name":"Schloss Dagstuhl \u2013 Leibniz Center for Informatics","host_organization_lineage":["https://openalex.org/I2799853480"],"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":"publishedVersion"},{"id":"doi:10.48550/arxiv.2001.02996","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2001.02996","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"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":"article"},{"id":"doi:10.4230/lipics.cpm.2026.6","is_oa":true,"landing_page_url":"https://doi.org/10.4230/lipics.cpm.2026.6","pdf_url":null,"source":{"id":"https://openalex.org/S7407052059","display_name":"Dagstuhl Research Online Publication Server","issn_l":null,"issn":[],"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":""}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2001.02996","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2001.02996","pdf_url":"https://arxiv.org/pdf/2001.02996","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"text"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":7,"referenced_works":["https://openalex.org/W2023034313","https://openalex.org/W2035314514","https://openalex.org/W2036042874","https://openalex.org/W2133943430","https://openalex.org/W2199929370","https://openalex.org/W2272542839","https://openalex.org/W2963469063"],"related_works":["https://openalex.org/W2068578356","https://openalex.org/W237537940","https://openalex.org/W1898422521","https://openalex.org/W2133478886","https://openalex.org/W2033995689","https://openalex.org/W4235762796","https://openalex.org/W1995096030","https://openalex.org/W2325476189","https://openalex.org/W4287212837","https://openalex.org/W2735923117"],"abstract_inverted_index":{"We":[0,45],"investigate":[1],"the":[2,12,27,48,61,70,134,139,154,162],"asymptotic":[3],"growth":[4],"of":[5,15,22,30,33,42,51,65,73,93,104,142,149,161,165,171],"function":[6],"CS(n),":[7],"which":[8,67],"maps":[9],"n":[10,24,37,86],"to":[11,145],"maximum":[13,28],"number":[14,29],"distinct":[16,31],"squares":[17,32,87,112],"in":[18,38,88,113],"a":[19,39,107,147],"circular":[20,151],"word":[21,40],"length":[23,34,43,94],"(that":[25],"is,":[26],"at":[35],"most":[36],"ww":[41],"2n).":[44],"improve":[46],"upon":[47],"lower":[49,135],"bound":[50,64,103,110],"1.25n":[52],"established":[53],"by":[54],"Amit":[55,97,172],"and":[56,60,75,98,124,173],"Gawrychowski":[57,99],"[SPIRE":[58],"2017]":[59],"straightforward":[62],"upper":[63,102,109,155],"2n,":[66],"follows":[68],"from":[69],"recent":[71],"result":[72],"Brlek":[74],"Li":[76],"[Comb.":[77],"Theory,":[78],"2025]":[79],"stating":[80],"that":[81,119],"there":[82],"are":[83],"fewer":[84],"than":[85],"standard":[89,114],"(i.e.,":[90],"non-circular)":[91],"words":[92,144],"n.":[95],"(Previously,":[96],"gave":[100],"an":[101,169],"32/15n":[105],"using":[106],"weaker":[108],"on":[111],"words.)":[115],"Specifically,":[116],"we":[117,137,157],"show":[118],"CS(n)":[120,130],"\u2264":[121],"\u23081.8":[122],"n\u2309":[123],"that,":[125],"for":[126],"infinitely":[127],"many":[128],"n,":[129],"\u2265":[131],"1.5n-\ud835\udcaa(\u221an).":[132],"For":[133,153],"bound,":[136,156],"exploit":[138,158],"combinatorial":[140],"structure":[141],"Fibonacci":[143],"construct":[146],"family":[148],"square-rich":[150],"words.":[152],"density":[159],"properties":[160],"starting":[163],"positions":[164],"long":[166],"squares,":[167],"adapting":[168],"approach":[170],"Gawrychowski.":[174]},"counts_by_year":[{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":1}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2020-01-23T00:00:00"}
