{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T03:17:01Z","timestamp":1740107821696,"version":"3.37.3"},"reference-count":6,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2024,4,23]],"date-time":"2024-04-23T00:00:00Z","timestamp":1713830400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,4,23]],"date-time":"2024-04-23T00:00:00Z","timestamp":1713830400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100005711","name":"Universit\u00e4t 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                    <mml:mi>A<\/mml:mi>\n                        <mml:mo>|<\/mml:mo>\n                      <\/mml:mrow>\n                      <mml:mn>3<\/mml:mn>\n                    <\/mml:msup>\n                    <mml:mo>\/<\/mml:mo>\n                    <mml:mi>K<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> has a subset <jats:inline-formula><jats:alternatives><jats:tex-math>$$A'\\subseteq A$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msup>\n                      <mml:mi>A<\/mml:mi>\n                      <mml:mo>\u2032<\/mml:mo>\n                    <\/mml:msup>\n                    <mml:mo>\u2286<\/mml:mo>\n                    <mml:mi>A<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> of size <jats:inline-formula><jats:alternatives><jats:tex-math>$$|A'|\\ge (1-\\varepsilon )K^{-1\/2}|A|$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mrow>\n                      <mml:mo>|<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:msup>\n                      <mml:mi>A<\/mml:mi>\n                      <mml:mo>\u2032<\/mml:mo>\n                    <\/mml:msup>\n                    <mml:mrow>\n                      <mml:mo>|<\/mml:mo>\n                      <mml:mo>\u2265<\/mml:mo>\n                      <mml:mrow>\n                        <mml:mo>(<\/mml:mo>\n                        <mml:mn>1<\/mml:mn>\n                        <mml:mo>-<\/mml:mo>\n                        <mml:mi>\u03b5<\/mml:mi>\n                        <mml:mo>)<\/mml:mo>\n                      <\/mml:mrow>\n                    <\/mml:mrow>\n                    <mml:msup>\n                      <mml:mi>K<\/mml:mi>\n                      <mml:mrow>\n                        <mml:mo>-<\/mml:mo>\n                        <mml:mn>1<\/mml:mn>\n                        <mml:mo>\/<\/mml:mo>\n                        <mml:mn>2<\/mml:mn>\n                      <\/mml:mrow>\n                    <\/mml:msup>\n                    <mml:mrow>\n                      <mml:mo>|<\/mml:mo>\n                      <mml:mi>A<\/mml:mi>\n                      <mml:mo>|<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> such that <jats:inline-formula><jats:alternatives><jats:tex-math>$$|A'-A'|\\le O_\\varepsilon (K^{4}|A'|)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mrow>\n                      <mml:mo>|<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:msup>\n                      <mml:mi>A<\/mml:mi>\n                      <mml:mo>\u2032<\/mml:mo>\n                    <\/mml:msup>\n                    <mml:mo>-<\/mml:mo>\n                    <mml:msup>\n                      <mml:mi>A<\/mml:mi>\n                      <mml:mo>\u2032<\/mml:mo>\n                    <\/mml:msup>\n                    <mml:mrow>\n                      <mml:mo>|<\/mml:mo>\n                      <mml:mo>\u2264<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:msub>\n                      <mml:mi>O<\/mml:mi>\n                      <mml:mi>\u03b5<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:msup>\n                      <mml:mi>K<\/mml:mi>\n                      <mml:mn>4<\/mml:mn>\n                    <\/mml:msup>\n                    <mml:mrow>\n                      <mml:mo>|<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:msup>\n                      <mml:mi>A<\/mml:mi>\n                      <mml:mo>\u2032<\/mml:mo>\n                    <\/mml:msup>\n                    <mml:mrow>\n                      <mml:mo>|<\/mml:mo>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. This is, essentially, the largest structured set one can get in the Balog\u2013Szemer\u00e9di\u2013Gowers theorem.<\/jats:p>","DOI":"10.1007\/s00493-024-00092-5","type":"journal-article","created":{"date-parts":[[2024,4,23]],"date-time":"2024-04-23T10:02:05Z","timestamp":1713866525000},"page":"691-698","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Note on the Theorem of Balog, Szemer\u00e9di, and Gowers"],"prefix":"10.1007","volume":"44","author":[{"given":"Christian","family":"Reiher","sequence":"first","affiliation":[]},{"given":"Tomasz","family":"Schoen","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,4,23]]},"reference":[{"key":"92_CR1","doi-asserted-by":"publisher","unstructured":"Balog, A.: Many additive quadruples, Additive combinatorics, CRM Proc. Lecture Notes, vol. 43, Amer. Math. 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