Kan Jiang , Derong Kong , Wenxia Li , Zhiqiang Wang
{"title":"康托尔集平移中的有理点","authors":"Kan Jiang , Derong Kong , Wenxia Li , Zhiqiang Wang","doi":"10.1016/j.indag.2024.03.012","DOIUrl":null,"url":null,"abstract":"<div><p>Given two coprime integers <span><math><mrow><mi>p</mi><mo>≥</mo><mn>2</mn></mrow></math></span> and <span><math><mrow><mi>q</mi><mo>≥</mo><mn>3</mn></mrow></math></span>, let <span><math><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>⊂</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span> consist of all rational numbers which have a finite <span><math><mi>p</mi></math></span>-ary expansion, and let <span><span><span><math><mrow><mi>K</mi><mrow><mo>(</mo><mi>q</mi><mo>,</mo><mi>A</mi><mo>)</mo></mrow><mo>=</mo><mfenced><mrow><munderover><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>∞</mi></mrow></munderover><mfrac><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>i</mi></mrow></msup></mrow></mfrac><mo>:</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mi>A</mi><mspace></mspace><mo>∀</mo><mi>i</mi><mo>∈</mo><mi>N</mi></mrow></mfenced><mo>,</mo></mrow></math></span></span></span>where <span><math><mrow><mi>A</mi><mo>⊂</mo><mfenced><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>q</mi><mo>−</mo><mn>1</mn></mrow></mfenced></mrow></math></span> with cardinality <span><math><mrow><mn>1</mn><mo><</mo><mi>#</mi><mi>A</mi><mo><</mo><mi>q</mi></mrow></math></span>. In 2021 Schleischitz showed that <span><math><mrow><mi>#</mi><mrow><mo>(</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>∩</mo><mi>K</mi><mrow><mo>(</mo><mi>q</mi><mo>,</mo><mi>A</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo><</mo><mo>+</mo><mi>∞</mi></mrow></math></span>. In this paper we show that for any <span><math><mrow><mi>r</mi><mo>∈</mo><mi>Q</mi></mrow></math></span> and for any <span><math><mrow><mi>α</mi><mo>∈</mo><mi>R</mi></mrow></math></span>, <span><span><span><math><mrow><mi>#</mi><mrow><mo>(</mo><mrow><mrow><mo>(</mo><mi>r</mi><msub><mrow><mi>D</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>+</mo><mi>α</mi><mo>)</mo></mrow><mo>∩</mo><mi>K</mi><mrow><mo>(</mo><mi>q</mi><mo>,</mo><mi>A</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo><</mo><mo>+</mo><mi>∞</mi><mo>.</mo></mrow></math></span></span></span></p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 3","pages":"Pages 516-522"},"PeriodicalIF":0.5000,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Rational points in translations of the Cantor set\",\"authors\":\"Kan Jiang , Derong Kong , Wenxia Li , Zhiqiang Wang\",\"doi\":\"10.1016/j.indag.2024.03.012\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Given two coprime integers <span><math><mrow><mi>p</mi><mo>≥</mo><mn>2</mn></mrow></math></span> and <span><math><mrow><mi>q</mi><mo>≥</mo><mn>3</mn></mrow></math></span>, let <span><math><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>⊂</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span> consist of all rational numbers which have a finite <span><math><mi>p</mi></math></span>-ary expansion, and let <span><span><span><math><mrow><mi>K</mi><mrow><mo>(</mo><mi>q</mi><mo>,</mo><mi>A</mi><mo>)</mo></mrow><mo>=</mo><mfenced><mrow><munderover><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>∞</mi></mrow></munderover><mfrac><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>i</mi></mrow></msup></mrow></mfrac><mo>:</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mi>A</mi><mspace></mspace><mo>∀</mo><mi>i</mi><mo>∈</mo><mi>N</mi></mrow></mfenced><mo>,</mo></mrow></math></span></span></span>where <span><math><mrow><mi>A</mi><mo>⊂</mo><mfenced><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>q</mi><mo>−</mo><mn>1</mn></mrow></mfenced></mrow></math></span> with cardinality <span><math><mrow><mn>1</mn><mo><</mo><mi>#</mi><mi>A</mi><mo><</mo><mi>q</mi></mrow></math></span>. In 2021 Schleischitz showed that <span><math><mrow><mi>#</mi><mrow><mo>(</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>∩</mo><mi>K</mi><mrow><mo>(</mo><mi>q</mi><mo>,</mo><mi>A</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo><</mo><mo>+</mo><mi>∞</mi></mrow></math></span>. In this paper we show that for any <span><math><mrow><mi>r</mi><mo>∈</mo><mi>Q</mi></mrow></math></span> and for any <span><math><mrow><mi>α</mi><mo>∈</mo><mi>R</mi></mrow></math></span>, <span><span><span><math><mrow><mi>#</mi><mrow><mo>(</mo><mrow><mrow><mo>(</mo><mi>r</mi><msub><mrow><mi>D</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>+</mo><mi>α</mi><mo>)</mo></mrow><mo>∩</mo><mi>K</mi><mrow><mo>(</mo><mi>q</mi><mo>,</mo><mi>A</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo><</mo><mo>+</mo><mi>∞</mi><mo>.</mo></mrow></math></span></span></span></p></div>\",\"PeriodicalId\":56126,\"journal\":{\"name\":\"Indagationes Mathematicae-New Series\",\"volume\":\"35 3\",\"pages\":\"Pages 516-522\"},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2024-05-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Indagationes Mathematicae-New Series\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0019357724000284\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Indagationes Mathematicae-New Series","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0019357724000284","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Given two coprime integers and , let consist of all rational numbers which have a finite -ary expansion, and let where with cardinality . In 2021 Schleischitz showed that . In this paper we show that for any and for any ,
期刊介绍：
Indagationes Mathematicae is a peer-reviewed international journal for the Mathematical Sciences of the Royal Dutch Mathematical Society. The journal aims at the publication of original mathematical research papers of high quality and of interest to a large segment of the mathematics community. The journal also welcomes the submission of review papers of high quality.
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