{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:22:50Z","timestamp":1776784970504,"version":"3.51.2"},"reference-count":7,"publisher":"American Mathematical Society (AMS)","issue":"253","license":[{"start":{"date-parts":[[2006,9,9]],"date-time":"2006-09-09T00:00:00Z","timestamp":1157760000000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    There is a 1941 conjecture of Erd\u0151s and Tur\u00e1n on what is now called additive basis that we restate:\n                    <bold>Conjecture 0.1<\/bold>\n                    (Erd\u0151s and Tur\u00e1n)\n                    <bold>.<\/bold>\n                    Suppose that\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"0 equals delta 0 greater-than delta 1 greater-than delta 2 greater-than delta 3 midline-horizontal-ellipsis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03b4\n                                \n                              <\/mml:mi>\n                              <mml:mn>0<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03b4\n                                \n                              <\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03b4\n                                \n                              <\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03b4\n                                \n                              <\/mml:mi>\n                              <mml:mn>3<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>\n                              \u22ef\n                              \n                            <\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">0 = \\delta _0&gt;\\delta _1&gt;\\delta _2&gt;\\delta _3\\cdots<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is an increasing sequence of integers and\n                    <disp-formula content-type=\"math\/mathml\">\n                      \\[\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"s left-parenthesis z right-parenthesis colon equals sigma-summation Underscript i equals 0 Overscript normal infinity Endscripts z Superscript delta Super Subscript i Superscript Baseline period\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>s<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>z<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>:=<\/mml:mo>\n                            <mml:munderover>\n                              <mml:mo>\n                                \u2211\n                                \n                              <\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>i<\/mml:mi>\n                                <mml:mo>=<\/mml:mo>\n                                <mml:mn>0<\/mml:mn>\n                              <\/mml:mrow>\n                              <mml:mi mathvariant=\"normal\">\n                                \u221e\n                                \n                              <\/mml:mi>\n                            <\/mml:munderover>\n                            <mml:msup>\n                              <mml:mi>z<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:msub>\n                                  <mml:mi>\n                                    \u03b4\n                                    \n                                  <\/mml:mi>\n                                  <mml:mi>i<\/mml:mi>\n                                <\/mml:msub>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo>.<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">s(z) : = \\sum _{i=0}^\\infty z^{\\delta _i}.<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                      \\]\n                    <\/disp-formula>\n                    Suppose that\n                    <disp-formula content-type=\"math\/mathml\">\n                      \\[\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"s squared left-parenthesis z right-parenthesis colon equals sigma-summation Underscript i equals 0 Overscript normal infinity Endscripts b Subscript i Baseline z Superscript i Baseline period\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mi>s<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>z<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>:=<\/mml:mo>\n                            <mml:munderover>\n                              <mml:mo>\n                                \u2211\n                                \n                              <\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>i<\/mml:mi>\n                                <mml:mo>=<\/mml:mo>\n                                <mml:mn>0<\/mml:mn>\n                              <\/mml:mrow>\n                              <mml:mi mathvariant=\"normal\">\n                                \u221e\n                                \n                              <\/mml:mi>\n                            <\/mml:munderover>\n                            <mml:msub>\n                              <mml:mi>b<\/mml:mi>\n                              <mml:mi>i<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:msup>\n                              <mml:mi>z<\/mml:mi>\n                              <mml:mi>i<\/mml:mi>\n                            <\/mml:msup>\n                            <mml:mo>.<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">s^2(z) := \\sum _{i=0}^\\infty b_i z^i.<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                      \\]\n                    <\/disp-formula>\n                    If\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"b Subscript i Baseline greater-than 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>b<\/mml:mi>\n                              <mml:mi>i<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">b_i&gt;0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for all\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"i\">\n                        <mml:semantics>\n                          <mml:mi>i<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">i<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , then\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-brace b Subscript n Baseline right-brace\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo fence=\"false\" stretchy=\"false\">{<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>b<\/mml:mi>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo fence=\"false\" stretchy=\"false\">}<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\{b_n\\}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is unbounded.  Our main purpose is to show that the sequence\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-brace b Subscript n Baseline right-brace\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo fence=\"false\" stretchy=\"false\">{<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>b<\/mml:mi>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo fence=\"false\" stretchy=\"false\">}<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\{b_n\\}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    cannot be bounded by\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"7\">\n                        <mml:semantics>\n                          <mml:mn>7<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">7<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . There is a surprisingly simple, though computationally very intensive, algorithm that establishes this.\n                  <\/p>","DOI":"10.1090\/s0025-5718-05-01777-1","type":"journal-article","created":{"date-parts":[[2005,11,16]],"date-time":"2005-11-16T10:22:35Z","timestamp":1132136555000},"page":"475-484","source":"Crossref","is-referenced-by-count":17,"title":["An old conjecture of Erdos\u2013Tur\u00e1n on additive bases"],"prefix":"10.1090","volume":"75","author":[{"given":"Peter","family":"Borwein","sequence":"first","affiliation":[]},{"given":"Stephen","family":"Choi","sequence":"additional","affiliation":[]},{"given":"Frank","family":"Chu","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2005,9,9]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"142","DOI":"10.1137\/0401016","article-title":"Questions related to the Erd\u0151s-Tur\u00e1n conjecture","volume":"1","author":"Dowd, Martin","year":"1988","journal-title":"SIAM J. Discrete Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0895-4801","issn-type":"print"},{"key":"2","unstructured":"P. Erd\u0151s and R. Frued, On Sidon-sequences and related problems, Mat. Lapok (New Ser.) (1991\/2 (in Hungarian)), no. 1, 1\u201344."},{"key":"3","doi-asserted-by":"publisher","first-page":"212","DOI":"10.1112\/jlms\/s1-16.4.212","article-title":"On a problem of Sidon in additive number theory, and on some related problems","volume":"16","author":"Erd\u00f6s, P.","year":"1941","journal-title":"J. London Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6107","issn-type":"print"},{"issue":"3-4","key":"4","first-page":"325","article-title":"Old and new problems and results in combinatorial number theory: van der Waerden\u2019s theorem and related topics","volume":"25","author":"Erd\u0151s, P.","year":"1979","journal-title":"Enseign. Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0013-8584","issn-type":"print"},{"issue":"2","key":"5","doi-asserted-by":"publisher","first-page":"339","DOI":"10.1016\/S0022-314X(03)00108-2","article-title":"On the Erd\u0151s-Tur\u00e1n conjecture","volume":"102","author":"Grekos, G.","year":"2003","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"issue":"1","key":"6","doi-asserted-by":"publisher","first-page":"1","DOI":"10.4064\/aa108-1-1","article-title":"Unique representation bases for the integers","volume":"108","author":"Nathanson, Melvyn B.","year":"2003","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"issue":"1-2","key":"7","doi-asserted-by":"publisher","first-page":"169","DOI":"10.1023\/A:1015261010544","article-title":"Range of bounded additive representation functions","volume":"42","author":"S\u00e1ndor, Csaba","year":"2001","journal-title":"Period. Math. Hungar.","ISSN":"https:\/\/id.crossref.org\/issn\/0031-5303","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2006-75-253\/S0025-5718-05-01777-1\/S0025-5718-05-01777-1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2006-75-253\/S0025-5718-05-01777-1\/S0025-5718-05-01777-1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T14:34:20Z","timestamp":1776782060000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2006-75-253\/S0025-5718-05-01777-1\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,9,9]]},"references-count":7,"journal-issue":{"issue":"253","published-print":{"date-parts":[[2006,1]]}},"alternative-id":["S0025-5718-05-01777-1"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-05-01777-1","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2005,9,9]]}}}