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Codes Cryptogr."],"published-print":{"date-parts":[[2023,4]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>For integers <jats:italic>m<\/jats:italic> and <jats:italic>n<\/jats:italic>, we study the problem of finding good lower bounds for the size of progression-free sets in <jats:inline-formula><jats:alternatives><jats:tex-math>$$(\\mathbb {Z}_{m}^{n},+)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:msubsup>\n                      <mml:mi>Z<\/mml:mi>\n                      <mml:mrow>\n                        <mml:mi>m<\/mml:mi>\n                      <\/mml:mrow>\n                      <mml:mi>n<\/mml:mi>\n                    <\/mml:msubsup>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mo>+<\/mml:mo>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Let <jats:inline-formula><jats:alternatives><jats:tex-math>$$r_{k}(\\mathbb {Z}_{m}^{n})$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msub>\n                      <mml:mi>r<\/mml:mi>\n                      <mml:mi>k<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:msubsup>\n                        <mml:mi>Z<\/mml:mi>\n                        <mml:mrow>\n                          <mml:mi>m<\/mml:mi>\n                        <\/mml:mrow>\n                        <mml:mi>n<\/mml:mi>\n                      <\/mml:msubsup>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> denote the maximal size of a subset of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathbb {Z}_{m}^{n}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msubsup>\n                    <mml:mi>Z<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mi>m<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mi>n<\/mml:mi>\n                  <\/mml:msubsup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> without arithmetic progressions of length\u00a0<jats:italic>k<\/jats:italic> and let <jats:inline-formula><jats:alternatives><jats:tex-math>$$P^{-}(m)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msup>\n                      <mml:mi>P<\/mml:mi>\n                      <mml:mo>-<\/mml:mo>\n                    <\/mml:msup>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>m<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> denote the least prime factor of\u00a0<jats:italic>m<\/jats:italic>. We construct explicit progression-free sets and obtain the following improved lower bounds for <jats:inline-formula><jats:alternatives><jats:tex-math>$$r_{k}(\\mathbb {Z}_{m}^{n})$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msub>\n                      <mml:mi>r<\/mml:mi>\n                      <mml:mi>k<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:msubsup>\n                        <mml:mi>Z<\/mml:mi>\n                        <mml:mrow>\n                          <mml:mi>m<\/mml:mi>\n                        <\/mml:mrow>\n                        <mml:mi>n<\/mml:mi>\n                      <\/mml:msubsup>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>:<jats:list list-type=\"bullet\">\n                <jats:list-item>\n                  <jats:p>If <jats:inline-formula><jats:alternatives><jats:tex-math>$$k\\ge 5$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <mml:mrow>\n                        <mml:mi>k<\/mml:mi>\n                        <mml:mo>\u2265<\/mml:mo>\n                        <mml:mn>5<\/mml:mn>\n                      <\/mml:mrow>\n                    <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is odd and <jats:inline-formula><jats:alternatives><jats:tex-math>$$P^{-}(m)\\ge (k+2)\/2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <mml:mrow>\n                        <mml:msup>\n                          <mml:mi>P<\/mml:mi>\n                          <mml:mo>-<\/mml:mo>\n                        <\/mml:msup>\n                        <mml:mrow>\n                          <mml:mo>(<\/mml:mo>\n                          <mml:mi>m<\/mml:mi>\n                          <mml:mo>)<\/mml:mo>\n                        <\/mml:mrow>\n                        <mml:mo>\u2265<\/mml:mo>\n                        <mml:mrow>\n                          <mml:mo>(<\/mml:mo>\n                          <mml:mi>k<\/mml:mi>\n                          <mml:mo>+<\/mml:mo>\n                          <mml:mn>2<\/mml:mn>\n                          <mml:mo>)<\/mml:mo>\n                        <\/mml:mrow>\n                        <mml:mo>\/<\/mml:mo>\n                        <mml:mn>2<\/mml:mn>\n                      <\/mml:mrow>\n                    <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, then <jats:disp-formula><jats:alternatives><jats:tex-math>$$\\begin{aligned} r_k(\\mathbb {Z}_m^n) \\gg _{m,k} \\frac{\\bigl \\lfloor \\frac{k-1}{k+1}m +1\\bigr \\rfloor ^{n}}{n^{\\lfloor \\frac{k-1}{k+1}m \\rfloor \/2}}. \\end{aligned}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <mml:mrow>\n                        <mml:mtable>\n                          <mml:mtr>\n                            <mml:mtd>\n                              <mml:mrow>\n                                <mml:msub>\n                                  <mml:mi>r<\/mml:mi>\n                                  <mml:mi>k<\/mml:mi>\n                                <\/mml:msub>\n                                <mml:mrow>\n                                  <mml:mo>(<\/mml:mo>\n                                  <mml:msubsup>\n                                    <mml:mi>Z<\/mml:mi>\n                                    <mml:mi>m<\/mml:mi>\n                                    <mml:mi>n<\/mml:mi>\n                                  <\/mml:msubsup>\n                                  <mml:mo>)<\/mml:mo>\n                                <\/mml:mrow>\n                                <mml:msub>\n                                  <mml:mo>\u226b<\/mml:mo>\n                                  <mml:mrow>\n                                    <mml:mi>m<\/mml:mi>\n                                    <mml:mo>,<\/mml:mo>\n                                    <mml:mi>k<\/mml:mi>\n                                  <\/mml:mrow>\n                                <\/mml:msub>\n                                <mml:mfrac>\n                                  <mml:mrow>\n                                    <mml:mrow>\n                                      <mml:mo>\u230a<\/mml:mo>\n                                    <\/mml:mrow>\n                                    <mml:mfrac>\n                                      <mml:mrow>\n                                        <mml:mi>k<\/mml:mi>\n                                        <mml:mo>-<\/mml:mo>\n                                        <mml:mn>1<\/mml:mn>\n                                      <\/mml:mrow>\n                                      <mml:mrow>\n                                        <mml:mi>k<\/mml:mi>\n                                        <mml:mo>+<\/mml:mo>\n                                        <mml:mn>1<\/mml:mn>\n                                      <\/mml:mrow>\n                                    <\/mml:mfrac>\n                                    <mml:mi>m<\/mml:mi>\n                                    <mml:mo>+<\/mml:mo>\n                                    <mml:mn>1<\/mml:mn>\n                                    <mml:msup>\n                                      <mml:mrow>\n                                        <mml:mo>\u230b<\/mml:mo>\n                                      <\/mml:mrow>\n                                      <mml:mi>n<\/mml:mi>\n                                    <\/mml:msup>\n                                  <\/mml:mrow>\n                                  <mml:msup>\n                                    <mml:mi>n<\/mml:mi>\n                                    <mml:mrow>\n                                      <mml:mo>\u230a<\/mml:mo>\n                                      <mml:mfrac>\n                                        <mml:mrow>\n                                          <mml:mi>k<\/mml:mi>\n                                          <mml:mo>-<\/mml:mo>\n                                          <mml:mn>1<\/mml:mn>\n                                        <\/mml:mrow>\n                                        <mml:mrow>\n                                          <mml:mi>k<\/mml:mi>\n                                          <mml:mo>+<\/mml:mo>\n                                          <mml:mn>1<\/mml:mn>\n                                        <\/mml:mrow>\n                                      <\/mml:mfrac>\n                                      <mml:mi>m<\/mml:mi>\n                                      <mml:mo>\u230b<\/mml:mo>\n                                      <mml:mo>\/<\/mml:mo>\n                                      <mml:mn>2<\/mml:mn>\n                                    <\/mml:mrow>\n                                  <\/mml:msup>\n                                <\/mml:mfrac>\n                                <mml:mo>.<\/mml:mo>\n                              <\/mml:mrow>\n                            <\/mml:mtd>\n                          <\/mml:mtr>\n                        <\/mml:mtable>\n                      <\/mml:mrow>\n                    <\/mml:math><\/jats:alternatives><\/jats:disp-formula><\/jats:p>\n                <\/jats:list-item>\n                <jats:list-item>\n                  <jats:p>If <jats:inline-formula><jats:alternatives><jats:tex-math>$$k\\ge 4$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <mml:mrow>\n                        <mml:mi>k<\/mml:mi>\n                        <mml:mo>\u2265<\/mml:mo>\n                        <mml:mn>4<\/mml:mn>\n                      <\/mml:mrow>\n                    <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is even, <jats:inline-formula><jats:alternatives><jats:tex-math>$$P^{-}(m) \\ge k$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <mml:mrow>\n                        <mml:msup>\n                          <mml:mi>P<\/mml:mi>\n                          <mml:mo>-<\/mml:mo>\n                        <\/mml:msup>\n                        <mml:mrow>\n                          <mml:mo>(<\/mml:mo>\n                          <mml:mi>m<\/mml:mi>\n                          <mml:mo>)<\/mml:mo>\n                        <\/mml:mrow>\n                        <mml:mo>\u2265<\/mml:mo>\n                        <mml:mi>k<\/mml:mi>\n                      <\/mml:mrow>\n                    <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$m \\equiv -1 \\bmod k$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <mml:mrow>\n                        <mml:mi>m<\/mml:mi>\n                        <mml:mo>\u2261<\/mml:mo>\n                        <mml:mo>-<\/mml:mo>\n                        <mml:mn>1<\/mml:mn>\n                        <mml:mspace\/>\n                        <mml:mo>mod<\/mml:mo>\n                        <mml:mspace\/>\n                        <mml:mi>k<\/mml:mi>\n                      <\/mml:mrow>\n                    <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, then <jats:disp-formula><jats:alternatives><jats:tex-math>$$\\begin{aligned} r_{k}(\\mathbb {Z}_{m}^{n}) \\gg _{m,k} \\frac{\\bigl \\lfloor \\frac{k-2}{k}m + 2\\bigr \\rfloor ^{n}}{n^{\\lfloor \\frac{k-2}{k}m + 1\\rfloor \/2}}. \\end{aligned}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <mml:mrow>\n                        <mml:mtable>\n                          <mml:mtr>\n                            <mml:mtd>\n                              <mml:mrow>\n                                <mml:msub>\n                                  <mml:mi>r<\/mml:mi>\n                                  <mml:mi>k<\/mml:mi>\n                                <\/mml:msub>\n                                <mml:mrow>\n                                  <mml:mo>(<\/mml:mo>\n                                  <mml:msubsup>\n                                    <mml:mi>Z<\/mml:mi>\n                                    <mml:mrow>\n                                      <mml:mi>m<\/mml:mi>\n                                    <\/mml:mrow>\n                                    <mml:mi>n<\/mml:mi>\n                                  <\/mml:msubsup>\n                                  <mml:mo>)<\/mml:mo>\n                                <\/mml:mrow>\n                                <mml:msub>\n                                  <mml:mo>\u226b<\/mml:mo>\n                                  <mml:mrow>\n                                    <mml:mi>m<\/mml:mi>\n                                    <mml:mo>,<\/mml:mo>\n                                    <mml:mi>k<\/mml:mi>\n                                  <\/mml:mrow>\n                                <\/mml:msub>\n                                <mml:mfrac>\n                                  <mml:mrow>\n                                    <mml:mrow>\n                                      <mml:mo>\u230a<\/mml:mo>\n                                    <\/mml:mrow>\n                                    <mml:mfrac>\n                                      <mml:mrow>\n                                        <mml:mi>k<\/mml:mi>\n                                        <mml:mo>-<\/mml:mo>\n                                        <mml:mn>2<\/mml:mn>\n                                      <\/mml:mrow>\n                                      <mml:mi>k<\/mml:mi>\n                                    <\/mml:mfrac>\n                                    <mml:mi>m<\/mml:mi>\n                                    <mml:mo>+<\/mml:mo>\n                                    <mml:mn>2<\/mml:mn>\n                                    <mml:msup>\n                                      <mml:mrow>\n                                        <mml:mo>\u230b<\/mml:mo>\n                                      <\/mml:mrow>\n                                      <mml:mi>n<\/mml:mi>\n                                    <\/mml:msup>\n                                  <\/mml:mrow>\n                                  <mml:msup>\n                                    <mml:mi>n<\/mml:mi>\n                                    <mml:mrow>\n                                      <mml:mo>\u230a<\/mml:mo>\n                                      <mml:mfrac>\n                                        <mml:mrow>\n                                          <mml:mi>k<\/mml:mi>\n                                          <mml:mo>-<\/mml:mo>\n                                          <mml:mn>2<\/mml:mn>\n                                        <\/mml:mrow>\n                                        <mml:mi>k<\/mml:mi>\n                                      <\/mml:mfrac>\n                                      <mml:mi>m<\/mml:mi>\n                                      <mml:mo>+<\/mml:mo>\n                                      <mml:mn>1<\/mml:mn>\n                                      <mml:mo>\u230b<\/mml:mo>\n                                      <mml:mo>\/<\/mml:mo>\n                                      <mml:mn>2<\/mml:mn>\n                                    <\/mml:mrow>\n                                  <\/mml:msup>\n                                <\/mml:mfrac>\n                                <mml:mo>.<\/mml:mo>\n                              <\/mml:mrow>\n                            <\/mml:mtd>\n                          <\/mml:mtr>\n                        <\/mml:mtable>\n                      <\/mml:mrow>\n                    <\/mml:math><\/jats:alternatives><\/jats:disp-formula><\/jats:p>\n                <\/jats:list-item>\n              <\/jats:list> Moreover, we give some further improved lower bounds on <jats:inline-formula><jats:alternatives><jats:tex-math>$$r_k(\\mathbb {Z}_p^n)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msub>\n                      <mml:mi>r<\/mml:mi>\n                      <mml:mi>k<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:msubsup>\n                        <mml:mi>Z<\/mml:mi>\n                        <mml:mi>p<\/mml:mi>\n                        <mml:mi>n<\/mml:mi>\n                      <\/mml:msubsup>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for primes <jats:inline-formula><jats:alternatives><jats:tex-math>$$p \\le 31$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>p<\/mml:mi>\n                    <mml:mo>\u2264<\/mml:mo>\n                    <mml:mn>31<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and progression lengths <jats:inline-formula><jats:alternatives><jats:tex-math>$$4 \\le k \\le 8$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mn>4<\/mml:mn>\n                    <mml:mo>\u2264<\/mml:mo>\n                    <mml:mi>k<\/mml:mi>\n                    <mml:mo>\u2264<\/mml:mo>\n                    <mml:mn>8<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s10623-022-01145-w","type":"journal-article","created":{"date-parts":[[2022,12,15]],"date-time":"2022-12-15T03:07:42Z","timestamp":1671073662000},"page":"1443-1452","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Large subsets of\u00a0$$\\mathbb {Z}_m^n$$ without arithmetic progressions"],"prefix":"10.1007","volume":"91","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2960-4030","authenticated-orcid":false,"given":"Christian","family":"Elsholtz","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Benjamin","family":"Klahn","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4362-429X","authenticated-orcid":false,"given":"Gabriel F.","family":"Lipnik","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,12,15]]},"reference":[{"key":"1145_CR1","doi-asserted-by":"publisher","first-page":"331","DOI":"10.1073\/pnas.32.12.331","volume":"32","author":"FA Behrend","year":"1946","unstructured":"Behrend F.A.: On sets of integers which contain no three terms in arithmetical progression. Proc. Natl Acad. Sci. U.S.A. 32, 331\u2013332 (1946). https:\/\/doi.org\/10.1073\/pnas.32.12.331.","journal-title":"Proc. Natl Acad. Sci. U.S.A."},{"issue":"1","key":"1145_CR2","doi-asserted-by":"publisher","first-page":"5","DOI":"10.1023\/A:1027365901231","volume":"31","author":"Y Edel","year":"2004","unstructured":"Edel Y.: Extensions of generalized product caps. Des. Codes Cryptogr. 31(1), 5\u201314 (2004). https:\/\/doi.org\/10.1023\/A:1027365901231.","journal-title":"Des. Codes Cryptogr."},{"issue":"1","key":"1145_CR3","doi-asserted-by":"publisher","first-page":"339","DOI":"10.4007\/annals.2017.185.1.8","volume":"185","author":"JS Ellenberg","year":"2017","unstructured":"Ellenberg J.S., Gijswijt D.: On large subsets of $${\\mathbb{F} }^n_q$$ with no three-term arithmetic progression. Ann. Math. (2) 185(1), 339\u2013343 (2017). https:\/\/doi.org\/10.4007\/annals.2017.185.1.8.","journal-title":"Ann. Math. (2)"},{"key":"1145_CR4","unstructured":"Elsholtz C., Lipnik G.F.: Exponentially larger affine and projective caps, submitted (2020)."},{"key":"1145_CR5","doi-asserted-by":"publisher","first-page":"2133","DOI":"10.1007\/s10623-020-00769-0","volume":"88","author":"C Elsholtz","year":"2020","unstructured":"Elsholtz C., Pach P.P.: Caps and progression-free sets in $$\\mathbb{Z} _m^n$$. Des. Codes Cryptogr. 88, 2133\u20132170 (2020). https:\/\/doi.org\/10.1007\/s10623-020-00769-0.","journal-title":"Des. Codes Cryptogr."},{"key":"1145_CR6","unstructured":"Garey M.R., Johnson D.S.: Computers and Intractability. W. H. Freeman and Co., San Francisco (1979) (A Guide to the Theory of NP-Completeness. 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