{"id":"https://openalex.org/W1682082215","doi":"https://doi.org/10.1090/mcom/2979","title":"On sets of integers which contain no three terms in geometric progression","display_name":"On sets of integers which contain no three terms in geometric progression","publication_year":2015,"publication_date":"2015-05-14","ids":{"openalex":"https://openalex.org/W1682082215","doi":"https://doi.org/10.1090/mcom/2979","mag":"1682082215"},"language":"en","primary_location":{"id":"doi:10.1090/mcom/2979","is_oa":true,"landing_page_url":"https://doi.org/10.1090/mcom/2979","pdf_url":"https://www.ams.org/mcom/2015-84-296/S0025-5718-2015-02979-2/S0025-5718-2015-02979-2.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://www.ams.org/mcom/2015-84-296/S0025-5718-2015-02979-2/S0025-5718-2015-02979-2.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5028054487","display_name":"Nathan McNew","orcid":"https://orcid.org/0000-0003-3062-989X"},"institutions":[{"id":"https://openalex.org/I107672454","display_name":"Dartmouth College","ror":"https://ror.org/049s0rh22","country_code":"US","type":"education","lineage":["https://openalex.org/I107672454"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Nathan McNew","raw_affiliation_strings":["Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755","Department of Mathematics, Dartmouth College, Hanover, New Hampshire, 03755"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755","institution_ids":["https://openalex.org/I107672454"]},{"raw_affiliation_string":"Department of Mathematics, Dartmouth College, Hanover, New Hampshire, 03755","institution_ids":["https://openalex.org/I107672454"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5028054487"],"corresponding_institution_ids":["https://openalex.org/I107672454"],"apc_list":null,"apc_paid":null,"fwci":4.6069,"has_fulltext":true,"cited_by_count":11,"citation_normalized_percentile":{"value":0.94105427,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"84","issue":"296","first_page":"2893","last_page":"2910"},"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":0.9998999834060669,"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":0.9998999834060669,"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/T11151","display_name":"Advanced Topology and Set Theory","score":0.9965999722480774,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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":0.996399998664856,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.9032807350158691},{"id":"https://openalex.org/keywords/geometric-progression","display_name":"Geometric progression","score":0.8455648422241211},{"id":"https://openalex.org/keywords/modulo","display_name":"Modulo","score":0.770967960357666},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6329072117805481},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.6016111373901367},{"id":"https://openalex.org/keywords/arithmetic-progression","display_name":"Arithmetic progression","score":0.5668284296989441},{"id":"https://openalex.org/keywords/natural-density","display_name":"Natural density","score":0.5170760154724121},{"id":"https://openalex.org/keywords/natural-number","display_name":"Natural number","score":0.5125986933708191},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.5113233923912048},{"id":"https://openalex.org/keywords/term","display_name":"Term (time)","score":0.4897301197052002},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4798718988895416},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.08160257339477539}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.9032807350158691},{"id":"https://openalex.org/C14367171","wikidata":"https://www.wikidata.org/wiki/Q173523","display_name":"Geometric progression","level":2,"score":0.8455648422241211},{"id":"https://openalex.org/C54732982","wikidata":"https://www.wikidata.org/wiki/Q1415345","display_name":"Modulo","level":2,"score":0.770967960357666},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6329072117805481},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.6016111373901367},{"id":"https://openalex.org/C130498168","wikidata":"https://www.wikidata.org/wiki/Q170008","display_name":"Arithmetic progression","level":2,"score":0.5668284296989441},{"id":"https://openalex.org/C156491182","wikidata":"https://www.wikidata.org/wiki/Q752723","display_name":"Natural density","level":2,"score":0.5170760154724121},{"id":"https://openalex.org/C184264201","wikidata":"https://www.wikidata.org/wiki/Q21199","display_name":"Natural number","level":2,"score":0.5125986933708191},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.5113233923912048},{"id":"https://openalex.org/C61797465","wikidata":"https://www.wikidata.org/wiki/Q1188986","display_name":"Term (time)","level":2,"score":0.4897301197052002},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4798718988895416},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.08160257339477539},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"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}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1090/mcom/2979","is_oa":true,"landing_page_url":"https://doi.org/10.1090/mcom/2979","pdf_url":"https://www.ams.org/mcom/2015-84-296/S0025-5718-2015-02979-2/S0025-5718-2015-02979-2.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1090/mcom/2979","is_oa":true,"landing_page_url":"https://doi.org/10.1090/mcom/2979","pdf_url":"https://www.ams.org/mcom/2015-84-296/S0025-5718-2015-02979-2/S0025-5718-2015-02979-2.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1682082215.pdf","grobid_xml":"https://content.openalex.org/works/W1682082215.grobid-xml"},"referenced_works_count":22,"referenced_works":["https://openalex.org/W1488916590","https://openalex.org/W1493612669","https://openalex.org/W1977514877","https://openalex.org/W1984582452","https://openalex.org/W1996922551","https://openalex.org/W2014727454","https://openalex.org/W2021610198","https://openalex.org/W2041824078","https://openalex.org/W2068893962","https://openalex.org/W2072865951","https://openalex.org/W2108900323","https://openalex.org/W2114487569","https://openalex.org/W2141605918","https://openalex.org/W2143054611","https://openalex.org/W3013063768","https://openalex.org/W3013532470","https://openalex.org/W3100139723","https://openalex.org/W3175367423","https://openalex.org/W4214617728","https://openalex.org/W4302023877","https://openalex.org/W4388288416","https://openalex.org/W6718866609"],"related_works":["https://openalex.org/W4226173127","https://openalex.org/W3013063768","https://openalex.org/W1880779767","https://openalex.org/W4300771890","https://openalex.org/W1507520630","https://openalex.org/W3013945579","https://openalex.org/W2131538623","https://openalex.org/W2924691597","https://openalex.org/W1682082215","https://openalex.org/W2549734191"],"abstract_inverted_index":{"The":[0],"problem":[1],"of":[2,6,67,80,104,112,174,182],"looking":[3,39],"for":[4,63],"subsets":[5,41],"the":[7,64,96,107,175],"natural":[8],"numbers":[9],"which":[10],"contain":[11],"no":[12],"three-term":[13,43],"arithmetic":[14],"progressions":[15,121],"has":[16],"a":[17,49,78,84,102,113,178],"rich":[18],"history.":[19],"Roth\u2019s":[20,163],"theorem":[21,164],"famously":[22],"shows":[23],"that":[24,98,119,166],"any":[25,170],"such":[26,48,68],"subset":[27,50],"cannot":[28,168],"have":[29,59],"positive":[30,172],"upper":[31,61,65,86,110,128,133],"density.":[32],"In":[33],"contrast,":[34],"Rankin":[35],"in":[36,95,122,165],"1960":[37],"suggested":[38],"at":[40],"without":[42],"geometric":[44,120,197],"progressions,":[45],"and":[46,76],"constructed":[47],"with":[51,83],"density":[52,66,87,111],"about":[53],"0.719.":[54],"More":[55],"recently,":[56],"several":[57],"authors":[58],"found":[60],"bounds":[62],"sets.":[69],"We":[70,116],"significantly":[71],"improve":[72],"upon":[73],"these":[74],"bounds,":[75],"demonstrate":[77],"method":[79,100],"constructing":[81],"sets":[82],"greater":[85],"than":[88],"Rankin\u2019s":[89],"set.":[90,115],"This":[91],"construction":[92],"is":[93],"optimal":[94],"sense":[97],"our":[99],"gives":[101],"way":[103],"effectively":[105],"computing":[106],"greatest":[108],"possible":[109],"geometric-progression-free":[114],"also":[117],"show":[118],"<inline-formula":[123,183],"content-type=\"math/mathml\">":[124,184],"<mml:math":[125,185],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[126,186],"alttext=\"double-struck":[127],"Z":[129],"slash":[130],"n":[131],"double-struck":[132],"Z\">":[134],"<mml:semantics>":[135,188],"<mml:mrow>":[136],"<mml:mrow":[137,142,147],"class=\"MJX-TeXAtom-ORD\">":[138,143,148],"<mml:mi":[139,149],"mathvariant=\"double-struck\">Z</mml:mi>":[140,150],"</mml:mrow>":[141,145,151,152],"<mml:mo>/</mml:mo>":[144],"<mml:mi>n</mml:mi>":[146,189],"<mml:annotation":[153,190],"encoding=\"application/x-tex\">\\mathbb":[154],"{Z}/n\\mathbb":[155],"{Z}</mml:annotation>":[156],"</mml:semantics>":[157,192],"</mml:math>":[158,193],"</inline-formula>":[159,194],"behave":[160],"more":[161],"like":[162],"one":[167],"take":[169],"fixed":[171],"proportion":[173],"integers":[176],"modulo":[177],"sufficiently":[179],"large":[180],"value":[181],"alttext=\"n\">":[187],"encoding=\"application/x-tex\">n</mml:annotation>":[191],"while":[195],"avoiding":[196],"progressions.":[198]},"counts_by_year":[{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":2}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}