{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:52:18Z","timestamp":1776797538206,"version":"3.51.2"},"reference-count":20,"publisher":"American Mathematical Society (AMS)","issue":"296","license":[{"start":{"date-parts":[[2016,5,14]],"date-time":"2016-05-14T00:00:00Z","timestamp":1463184000000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    The problem of looking for subsets of the natural numbers which contain no three-term arithmetic progressions has a rich history. Roth\u2019s theorem famously shows that any such subset cannot have positive upper density. In contrast, Rankin in 1960 suggested looking at subsets without three-term geometric progressions, and constructed such a subset with density about 0.719. More recently, several authors have found upper bounds for the upper density of such sets. We significantly improve upon these bounds, and demonstrate a method of constructing sets with a greater upper density than Rankin\u2019s set. This construction is optimal in the sense that our method gives a way of effectively computing the greatest possible upper density of a geometric-progression-free set. We also show that geometric progressions in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper Z slash n double-struck upper Z\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbb {Z}\/n\\mathbb {Z}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    behave more like Roth\u2019s theorem in that one cannot take any fixed positive proportion of the integers modulo a sufficiently large value of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n\">\n                        <mml:semantics>\n                          <mml:mi>n<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    while avoiding geometric progressions.\n                  <\/p>","DOI":"10.1090\/mcom\/2979","type":"journal-article","created":{"date-parts":[[2015,5,14]],"date-time":"2015-05-14T14:41:35Z","timestamp":1431614495000},"page":"2893-2910","source":"Crossref","is-referenced-by-count":5,"title":["On sets of integers which contain no three terms in geometric progression"],"prefix":"10.1090","volume":"84","author":[{"given":"Nathan","family":"McNew","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2015,5,14]]},"reference":[{"issue":"7","key":"1","doi-asserted-by":"publisher","first-page":"1219","DOI":"10.1016\/j.jcta.2005.11.003","article-title":"Multiplicative structures in additively large sets","volume":"113","author":"Beiglb\u00f6ck, Mathias","year":"2006","journal-title":"J. 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