{"title":"线性形式的贪婪西顿集","authors":"Yin Choi Cheng","doi":"10.1016/j.jnt.2024.07.010","DOIUrl":null,"url":null,"abstract":"<div><p>The greedy Sidon set, also known as the Mian-Chowla sequence, is the lexicographically first set in <span><math><mi>N</mi></math></span> that does not contain <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> with <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. Its growth and structure have remained enigmatic for 80 years. In this work, we study a generalization from the form <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> to arbitrary linear forms <span><math><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mo>…</mo><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>h</mi></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mi>h</mi></mrow></msub></math></span>; these are called Sidon sets for linear forms. We explicitly describe the elements of the greedy Sidon sets for linear forms when <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> for some <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>, and also when <span><math><mi>h</mi><mo>=</mo><mn>2</mn><mo>,</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>2</mn><mo>,</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>≥</mo><mn>4</mn></math></span>, the “structured” domain. We also contrast the “enigmatic” domain when <span><math><mi>h</mi><mo>=</mo><mn>2</mn><mo>,</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>2</mn><mo>,</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mn>3</mn></math></span> with the “structured” domain, and give upper bounds on the growth rates in both cases.</p></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022314X24001768/pdfft?md5=530dddb3b9f53a0f7a336819d6924b12&pid=1-s2.0-S0022314X24001768-main.pdf","citationCount":"0","resultStr":"{\"title\":\"Greedy Sidon sets for linear forms\",\"authors\":\"Yin Choi Cheng\",\"doi\":\"10.1016/j.jnt.2024.07.010\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>The greedy Sidon set, also known as the Mian-Chowla sequence, is the lexicographically first set in <span><math><mi>N</mi></math></span> that does not contain <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> with <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. Its growth and structure have remained enigmatic for 80 years. In this work, we study a generalization from the form <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> to arbitrary linear forms <span><math><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mo>…</mo><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>h</mi></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mi>h</mi></mrow></msub></math></span>; these are called Sidon sets for linear forms. We explicitly describe the elements of the greedy Sidon sets for linear forms when <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> for some <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>, and also when <span><math><mi>h</mi><mo>=</mo><mn>2</mn><mo>,</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>2</mn><mo>,</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>≥</mo><mn>4</mn></math></span>, the “structured” domain. We also contrast the “enigmatic” domain when <span><math><mi>h</mi><mo>=</mo><mn>2</mn><mo>,</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>2</mn><mo>,</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mn>3</mn></math></span> with the “structured” domain, and give upper bounds on the growth rates in both cases.</p></div>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-08-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S0022314X24001768/pdfft?md5=530dddb3b9f53a0f7a336819d6924b12&pid=1-s2.0-S0022314X24001768-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0022314X24001768\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022314X24001768","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
The greedy Sidon set, also known as the Mian-Chowla sequence, is the lexicographically first set in that does not contain with . Its growth and structure have remained enigmatic for 80 years. In this work, we study a generalization from the form to arbitrary linear forms ; these are called Sidon sets for linear forms. We explicitly describe the elements of the greedy Sidon sets for linear forms when for some , and also when , the “structured” domain. We also contrast the “enigmatic” domain when with the “structured” domain, and give upper bounds on the growth rates in both cases.