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Codes Cryptogr."],"published-print":{"date-parts":[[2026,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    Given a finite abelian group\n                    <jats:italic>G<\/jats:italic>\n                    and a subset\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$J\\subset G$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>J<\/mml:mi>\n                            <mml:mo>\u2282<\/mml:mo>\n                            <mml:mi>G<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    with\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$0\\in J$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>\u2208<\/mml:mo>\n                            <mml:mi>J<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , let\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$D_{G}(J,N)$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>D<\/mml:mi>\n                              <mml:mi>G<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mi>J<\/mml:mi>\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>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    be the maximum size of\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$A\\subset G^{N}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>A<\/mml:mi>\n                            <mml:mo>\u2282<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>G<\/mml:mi>\n                              <mml:mi>N<\/mml:mi>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    such that the difference set\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$A-A$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>A<\/mml:mi>\n                            <mml:mo>-<\/mml:mo>\n                            <mml:mi>A<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$J^{N}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:msup>\n                            <mml:mi>J<\/mml:mi>\n                            <mml:mi>N<\/mml:mi>\n                          <\/mml:msup>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    have no non-trivial intersection. Recently, this extremal problem has been widely studied for different groups\n                    <jats:italic>G<\/jats:italic>\n                    and subsets\n                    <jats:italic>J<\/jats:italic>\n                    . In this paper, we generalize and improve the relevant results by Alon and by Heged\u0171s by building a bridge between this problem and cyclotomic polynomials with the help of algebraic graph theory. In particular, we construct infinitely many non-trivial families of\n                    <jats:italic>G<\/jats:italic>\n                    and\n                    <jats:italic>J<\/jats:italic>\n                    for which the current known upper bounds on\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$D_{G}(J, N)$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>D<\/mml:mi>\n                              <mml:mi>G<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mi>J<\/mml:mi>\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>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    can be improved exponentially.\n                  <\/jats:p>","DOI":"10.1007\/s10623-025-01760-3","type":"journal-article","created":{"date-parts":[[2026,1,5]],"date-time":"2026-01-05T05:11:40Z","timestamp":1767589900000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Intersective sets over abelian groups"],"prefix":"10.1007","volume":"94","author":[{"given":"Zixiang","family":"Xu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chi Hoi","family":"Yip","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2026,1,5]]},"reference":[{"key":"1760_CR1","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.jnt.2021.04.013","volume":"229","author":"A Al-Kateeb","year":"2021","unstructured":"Al-Kateeb A., Ambrosino M., Hong H., Lee E.: Maximum gap in cyclotomic polynomials. 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