{"id":"https://openalex.org/W4382793646","doi":"https://doi.org/10.1007/s00454-023-00523-y","title":"Perfectly Packing a Square by Squares of Nearly Harmonic Sidelength","display_name":"Perfectly Packing a Square by Squares of Nearly Harmonic Sidelength","publication_year":2023,"publication_date":"2023-07-01","ids":{"openalex":"https://openalex.org/W4382793646","doi":"https://doi.org/10.1007/s00454-023-00523-y","pmid":"https://pubmed.ncbi.nlm.nih.gov/38720937"},"language":"lv","primary_location":{"id":"doi:10.1007/s00454-023-00523-y","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00454-023-00523-y","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00454-023-00523-y.pdf","source":{"id":"https://openalex.org/S103671469","display_name":"Discrete & Computational Geometry","issn_l":"0179-5376","issn":["0179-5376","1432-0444"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Discrete &amp; Computational Geometry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","pubmed"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s00454-023-00523-y.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5066728260","display_name":"Terence Tao","orcid":"https://orcid.org/0000-0002-0140-7641"},"institutions":[{"id":"https://openalex.org/I161318765","display_name":"University of California, Los Angeles","ror":"https://ror.org/046rm7j60","country_code":"US","type":"education","lineage":["https://openalex.org/I161318765"]},{"id":"https://openalex.org/I2799798094","display_name":"UCLA Health","ror":"https://ror.org/01d88se56","country_code":"US","type":"healthcare","lineage":["https://openalex.org/I2799798094"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Terence Tao","raw_affiliation_strings":["UCLA Department of Mathematics, Los Angeles, CA 90095-1555 USA","UCLA Department of Mathematics, Los Angeles, CA, 90095-1555, USA"],"raw_orcid":"https://orcid.org/0000-0002-0140-7641","affiliations":[{"raw_affiliation_string":"UCLA Department of Mathematics, Los Angeles, CA 90095-1555 USA","institution_ids":["https://openalex.org/I161318765","https://openalex.org/I2799798094"]},{"raw_affiliation_string":"UCLA Department of Mathematics, Los Angeles, CA, 90095-1555, USA","institution_ids":["https://openalex.org/I161318765"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5066728260"],"corresponding_institution_ids":["https://openalex.org/I161318765","https://openalex.org/I2799798094"],"apc_list":{"value":2290,"currency":"EUR","value_usd":2890},"apc_paid":{"value":2290,"currency":"EUR","value_usd":2890},"fwci":4.6312,"has_fulltext":true,"cited_by_count":9,"citation_normalized_percentile":{"value":0.96533333,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":91,"max":97},"biblio":{"volume":"71","issue":"4","first_page":"1178","last_page":"1189"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9972000122070312,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9972000122070312,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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":0.9955999851226807,"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/T11797","display_name":"graph theory and CDMA systems","score":0.9902999997138977,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.4521791934967041},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.20649030804634094}],"concepts":[{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.4521791934967041},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.20649030804634094}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1007/s00454-023-00523-y","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00454-023-00523-y","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00454-023-00523-y.pdf","source":{"id":"https://openalex.org/S103671469","display_name":"Discrete & Computational Geometry","issn_l":"0179-5376","issn":["0179-5376","1432-0444"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Discrete &amp; Computational Geometry","raw_type":"journal-article"},{"id":"pmid:38720937","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/38720937","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Discrete & computational geometry","raw_type":null},{"id":"pmh:oai:escholarship.org:ark:/13030/qt0qj1j6ww","is_oa":true,"landing_page_url":"https://escholarship.org/uc/item/0qj1j6ww","pdf_url":"https://escholarship.org/content/qt0qj1j6ww/qt0qj1j6ww.pdf","source":{"id":"https://openalex.org/S4306400115","display_name":"eScholarship (California Digital Library)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I2801248553","host_organization_name":"California Digital Library","host_organization_lineage":["https://openalex.org/I2801248553"],"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":"Discrete & Computational Geometry, vol 71, iss 4","raw_type":"article"},{"id":"pmh:oai:pubmedcentral.nih.gov:11073991","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/11073991","pdf_url":"https://pmc.ncbi.nlm.nih.gov/articles/PMC11073991/pdf/454_2023_Article_523.pdf","source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"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":"Discrete Comput Geom","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.1007/s00454-023-00523-y","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00454-023-00523-y","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00454-023-00523-y.pdf","source":{"id":"https://openalex.org/S103671469","display_name":"Discrete & Computational Geometry","issn_l":"0179-5376","issn":["0179-5376","1432-0444"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Discrete &amp; Computational Geometry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G5323917965","display_name":null,"funder_award_id":"DMS-1764034","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G6417927605","display_name":null,"funder_award_id":"1764034","funder_id":"https://openalex.org/F4320332172","funder_display_name":"Directorate for Mathematical and Physical Sciences"},{"id":"https://openalex.org/G8533293066","display_name":null,"funder_award_id":"1764034","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"},{"id":"https://openalex.org/F4320306164","display_name":"Simons Foundation","ror":"https://ror.org/01cmst727"},{"id":"https://openalex.org/F4320332172","display_name":"Directorate for Mathematical and Physical Sciences","ror":"https://ror.org/029b7h395"}],"has_content":{"pdf":true,"grobid_xml":false},"content_urls":{"pdf":"https://content.openalex.org/works/W4382793646.pdf"},"referenced_works_count":14,"referenced_works":["https://openalex.org/W1539895965","https://openalex.org/W1541507112","https://openalex.org/W1966497983","https://openalex.org/W2032640868","https://openalex.org/W2040288558","https://openalex.org/W2045151914","https://openalex.org/W2810047227","https://openalex.org/W2887743469","https://openalex.org/W2899964893","https://openalex.org/W3129159089","https://openalex.org/W4295634057","https://openalex.org/W4313216067","https://openalex.org/W4320728553","https://openalex.org/W6604472716"],"related_works":["https://openalex.org/W2051487156","https://openalex.org/W2073681303","https://openalex.org/W2053286651","https://openalex.org/W2181743346","https://openalex.org/W2187401768","https://openalex.org/W2181413294","https://openalex.org/W2989452537","https://openalex.org/W2052122378","https://openalex.org/W2544423928","https://openalex.org/W2062023542"],"abstract_inverted_index":{"Abstract":[0],"A":[1],"well-known":[2],"open":[3],"problem":[4],"of":[5,13,35,114,148,189],"Meir":[6],"and":[7,94],"Moser":[8],"asks":[9],"if":[10],"the":[11,112],"squares":[12,113],"sidelength":[14,115],"1/":[15],"n":[16],"for":[17,78,127,192],"$$n\\ge":[18,128],"2$$":[19],"<mml:math":[20,41,81,97,117,130,153,195,216],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">":[21,42,82,98,118,131,154,196,217],"<mml:mrow>":[22,43,46,55,83,121,132,155,158,170,197,218],"<mml:mi>n</mml:mi>":[23,47,54,100,120,133,136,159,162,169,220],"<mml:mo>\u2265</mml:mo>":[24,134],"<mml:mn>2</mml:mn>":[25,49,57,63,86,172,200,204],"</mml:mrow>":[26,50,58,69,91,124,139,165,174,176,207,225],"</mml:math>":[27,70,92,103,126,140,177,208,226],"can":[28,141,213],"be":[29,142],"packed":[30,143],"perfectly":[31,144],"into":[32,145],"a":[33,146,186,190],"rectangle":[34,187],"area":[36,149],"$$\\sum":[37,150],"_{n=2}^\\infty":[38],"n^{-2}=\\pi":[39],"^2/6-1$$":[40],"<mml:msubsup>":[44,156],"<mml:mo>\u2211</mml:mo>":[45,157],"<mml:mo>=</mml:mo>":[48,60,160,223],"<mml:mi>\u221e</mml:mi>":[51,166],"</mml:msubsup>":[52,167],"<mml:msup>":[53,61,119,168],"<mml:mo>-</mml:mo>":[56,67,122,171],"</mml:msup>":[59,64,125,175],"<mml:mi>\u03c0</mml:mi>":[62],"<mml:mo>/</mml:mo>":[65,85,199,205],"<mml:mn>6</mml:mn>":[66],"<mml:mn>1</mml:mn>":[68,84,90,198,224],".":[71,178],"In":[72],"this":[73],"paper":[74],"we":[75],"show":[76],"that":[77,104],"any":[79,95],"$$1/2&lt;t&lt;1$$":[80],"<mml:mo>&lt;</mml:mo>":[87,89,201],"<mml:mi>t</mml:mi>":[88,123,173,202],",":[93,111],"$$n_0$$":[96],"<mml:msub>":[99,135,161,219],"<mml:mn>0</mml:mn>":[101,137,163,221],"</mml:msub>":[102,138,164,222],"is":[105],"sufficiently":[106],"large":[107],"depending":[108],"on":[109],"t":[110],"$$n^{-t}$$":[116],"n_0$$":[129],"square":[147],"_{n=n_0}^\\infty":[151],"n^{-2t}$$":[152],"This":[179],"was":[180],"previously":[181],"known":[182],"(if":[183],"one":[184,212],"packs":[185],"instead":[188],"square)":[191],"$$1/2&lt;t\\le":[193],"2/3$$":[194],"<mml:mo>\u2264</mml:mo>":[203],"<mml:mn>3</mml:mn>":[206],"(in":[209],"which":[210],"case":[211],"take":[214],"$$n_0=1$$":[215],").":[227]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":3},{"year":2022,"cited_by_count":2}],"updated_date":"2026-05-16T08:24:45.110214","created_date":"2025-10-10T00:00:00"}