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Anal."],"published-print":{"date-parts":[[2025,12]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    A set of integers\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$A$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>A<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    is\n                    <jats:italic>non-averaging<\/jats:italic>\n                    if there is no element\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$a$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>a<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    in\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$A$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>A<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    which can be written as an average of a subset of\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$A$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>A<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    not containing\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$a$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>a<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . We show that the largest non-averaging subset of\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$\\{1, \\ldots , n\\}$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mo>{<\/mml:mo>\n                          <mml:mn>1<\/mml:mn>\n                          <mml:mo>,<\/mml:mo>\n                          <mml:mo>\u2026<\/mml:mo>\n                          <mml:mo>,<\/mml:mo>\n                          <mml:mi>n<\/mml:mi>\n                          <mml:mo>}<\/mml:mo>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    has size\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$n^{1\/4+o(1)}$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:msup>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mrow>\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mo>\/<\/mml:mo>\n                              <mml:mn>4<\/mml:mn>\n                              <mml:mo>+<\/mml:mo>\n                              <mml:mi>o<\/mml:mi>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , thus solving the Erd\u0151s\u2013Straus problem. We also determine the largest size of a non-averaging set in a\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$d$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>d<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    -dimensional box for any fixed\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$d$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>d<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . Our main tool includes the structure theorem for the set of subset sums due to Conlon, Fox and the first author, together with a result about the structure of a point set in nearly convex position.\n                  <\/jats:p>","DOI":"10.1007\/s00039-025-00728-8","type":"journal-article","created":{"date-parts":[[2025,12,3]],"date-time":"2025-12-03T09:02:56Z","timestamp":1764752576000},"page":"1712-1738","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Sharp Bound for the Erd\u0151s\u2013Straus Non-averaging Set Problem"],"prefix":"10.1007","volume":"35","author":[{"given":"Huy Tuan","family":"Pham","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dmitrii","family":"Zakharov","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,12,3]]},"reference":[{"key":"728_CR1","series-title":"Congr. 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