{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T07:50:17Z","timestamp":1776844217776,"version":"3.51.2"},"reference-count":8,"publisher":"American Mathematical Society (AMS)","issue":"216","license":[{"start":{"date-parts":[[1997,10,1]],"date-time":"1997-10-01T00:00:00Z","timestamp":875664000000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    Several papers have investigated sequences which have no\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k\">\n                        <mml:semantics>\n                          <mml:mi>k<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -term arithmetic progressions, finding bounds on their density and looking at sequences generated by greedy algorithms. Rankin in 1960 suggested looking at sequences without\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k\">\n                        <mml:semantics>\n                          <mml:mi>k<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -term geometric progressions, and constructed such sequences for each\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k\">\n                        <mml:semantics>\n                          <mml:mi>k<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    with positive density. In this paper we improve on Rankin\u2019s results, derive upper bounds, and look at sequences generated by a greedy algorithm.\n                  <\/p>","DOI":"10.1090\/s0025-5718-96-00765-x","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:28Z","timestamp":1027707268000},"page":"1749-1754","source":"Crossref","is-referenced-by-count":4,"title":["On sequences without geometric progressions"],"prefix":"10.1090","volume":"65","author":[{"given":"Brienne","family":"Brown","sequence":"first","affiliation":[]},{"given":"Daniel","family":"Gordon","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1996,10,1]]},"reference":[{"key":"1","doi-asserted-by":"crossref","unstructured":"P. Erd\u0151s and P. Tur\u00e1n, On some sequences of integers, J. London Math. Soc. 11 (1936), 261\u2013264.","DOI":"10.1112\/jlms\/s1-11.4.261"},{"issue":"148","key":"2","doi-asserted-by":"publisher","first-page":"1353","DOI":"10.2307\/2006472","article-title":"Sets of integers with no long arithmetic progressions generated by the greedy algorithm","volume":"33","author":"Gerver, Joseph L.","year":"1979","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"3","key":"3","doi-asserted-by":"publisher","first-page":"765","DOI":"10.2307\/2047261","article-title":"Greedily partitioning the natural numbers into sets free of arithmetic progressions","volume":"102","author":"Gerver, Joseph","year":"1988","journal-title":"Proc. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9939","issn-type":"print"},{"key":"4","unstructured":"Richard K. Guy, Unsolved problems in number theory, second ed., Springer\u2013Verlag, 1994."},{"key":"5","unstructured":"A. M. Odlyzko and R. P. Stanley, Some curious sequences constructed with the greedy algorithm, Bell Labs internal memo, 1978."},{"key":"6","first-page":"332","article-title":"Sets of integers containing not more than a given number of terms in arithmetical progression","volume":"65","author":"Rankin, R. A.","year":"1960","journal-title":"Proc. Roy. Soc. Edinburgh Sect. A","ISSN":"https:\/\/id.crossref.org\/issn\/0308-2105","issn-type":"print"},{"key":"7","first-page":"285","article-title":"Sur les inverses des \u00e9l\u00e9ments d\u00e9rivables dans un anneau abstrait","volume":"209","author":"Hebroni, P.","year":"1939","journal-title":"C. R. Acad. Sci. Paris","ISSN":"https:\/\/id.crossref.org\/issn\/0001-4036","issn-type":"print"},{"key":"8","doi-asserted-by":"publisher","first-page":"199","DOI":"10.4064\/aa-27-1-199-245","article-title":"On sets of integers containing no \ud835\udc58 elements in arithmetic progression","volume":"27","author":"Szemer\u00e9di, E.","year":"1975","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/1996-65-216\/S0025-5718-96-00765-X\/S0025-5718-96-00765-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1996-65-216\/S0025-5718-96-00765-X\/S0025-5718-96-00765-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:17:52Z","timestamp":1776719872000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1996-65-216\/S0025-5718-96-00765-X\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996,10,1]]},"references-count":8,"journal-issue":{"issue":"216","published-print":{"date-parts":[[1996,10]]}},"alternative-id":["S0025-5718-96-00765-X"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-96-00765-x","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[1996,10,1]]}}}