{"title":"乘法互补，II.","authors":"","doi":"10.1016/j.jnt.2024.05.014","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper we prove that if <em>A</em> and <em>B</em> are infinite subsets of positive integers such that every positive integer <em>n</em> can be written as <span><math><mi>n</mi><mo>=</mo><mi>a</mi><mi>b</mi></math></span>, <span><math><mi>a</mi><mo>∈</mo><mi>A</mi></math></span>, <span><math><mi>b</mi><mo>∈</mo><mi>B</mi></math></span>, then <span><math><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mo>∞</mo></mrow></munder><mo></mo><mfrac><mrow><mi>A</mi><mo>(</mo><mi>x</mi><mo>)</mo><mi>B</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><mi>x</mi></mrow></mfrac><mo>=</mo><mo>∞</mo></math></span>. We present some tight density bounds in connection with multiplicative complements.</p></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Multiplicative complements, II.\",\"authors\":\"\",\"doi\":\"10.1016/j.jnt.2024.05.014\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this paper we prove that if <em>A</em> and <em>B</em> are infinite subsets of positive integers such that every positive integer <em>n</em> can be written as <span><math><mi>n</mi><mo>=</mo><mi>a</mi><mi>b</mi></math></span>, <span><math><mi>a</mi><mo>∈</mo><mi>A</mi></math></span>, <span><math><mi>b</mi><mo>∈</mo><mi>B</mi></math></span>, then <span><math><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mo>∞</mo></mrow></munder><mo></mo><mfrac><mrow><mi>A</mi><mo>(</mo><mi>x</mi><mo>)</mo><mi>B</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><mi>x</mi></mrow></mfrac><mo>=</mo><mo>∞</mo></math></span>. We present some tight density bounds in connection with multiplicative complements.</p></div>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-06-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0022314X24001458\",\"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/S0022314X24001458","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
In this paper we prove that if A and B are infinite subsets of positive integers such that every positive integer n can be written as , , , then . We present some tight density bounds in connection with multiplicative complements.
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