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It is known that <jats:italic>NA<\/jats:italic> has a particular size (<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\vert NA\\vert = P_A(N)$$<\/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:mi>N<\/mml:mi>\n                      <mml:mi>A<\/mml:mi>\n                      <mml:mo>|<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>P<\/mml:mi>\n                      <mml:mi>A<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>N<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for some <jats:inline-formula><jats:alternatives><jats:tex-math>$$P_A(X) \\in {\\mathbb {Q}}[X]$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msub>\n                      <mml:mi>P<\/mml:mi>\n                      <mml:mi>A<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>X<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>\u2208<\/mml:mo>\n                    <mml:mi>Q<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mo>[<\/mml:mo>\n                      <mml:mi>X<\/mml:mi>\n                      <mml:mo>]<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>) and structure (all of the lattice points in a cone other than certain exceptional sets), once <jats:italic>N<\/jats:italic> is larger than some threshold. In this article we give the first effective upper bounds for this threshold for arbitrary <jats:italic>A<\/jats:italic>. Such explicit results were only previously known in the special cases when <jats:inline-formula><jats:alternatives><jats:tex-math>$$d=1$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>d<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, when the convex hull of <jats:italic>A<\/jats:italic> is a simplex or when <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\vert A\\vert = d+2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>|<\/mml:mo>\n                    <mml:mi>A<\/mml:mi>\n                    <mml:mo>|<\/mml:mo>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mi>d<\/mml:mi>\n                    <mml:mo>+<\/mml:mo>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> Curran and Goldmakher (Discrete Anal. Paper No. 27, 2021), results which we improve.<\/jats:p>","DOI":"10.1007\/s00493-023-00055-2","type":"journal-article","created":{"date-parts":[[2023,9,18]],"date-time":"2023-09-18T12:02:11Z","timestamp":1695038531000},"page":"1139-1178","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Effective Results on the Size and Structure of Sumsets"],"prefix":"10.1007","volume":"43","author":[{"given":"Andrew","family":"Granville","sequence":"first","affiliation":[]},{"given":"George","family":"Shakan","sequence":"additional","affiliation":[]},{"given":"Aled","family":"Walker","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,9,18]]},"reference":[{"issue":"1","key":"55_CR1","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1007\/BF01393823","volume":"73","author":"E Bombieri","year":"1983","unstructured":"Bombieri, E., Vaaler, J.: On Siegel\u2019s lemma. 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