{"title":"Uniform Diophantine approximation with restricted denominators","authors":"Bo Wang , Bing Li , Ruofan Li","doi":"10.1016/j.jnt.2024.02.017","DOIUrl":null,"url":null,"abstract":"<div><p>Let <span><math><mi>b</mi><mo>≥</mo><mn>2</mn></math></span> be an integer and <span><math><mi>A</mi><mo>=</mo><msubsup><mrow><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></msubsup></math></span> be a strictly increasing subsequence of positive integers with <span><math><mi>η</mi><mo>:</mo><mo>=</mo><munder><mrow><mi>lim sup</mi></mrow><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></munder><mspace></mspace><mfrac><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></mfrac><mo><</mo><mo>+</mo><mo>∞</mo></math></span>. For each irrational real number <em>ξ</em>, we denote by <span><math><msub><mrow><mover><mrow><mi>v</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>b</mi><mo>,</mo><mi>A</mi></mrow></msub><mo>(</mo><mi>ξ</mi><mo>)</mo></math></span> the supremum of the real numbers <span><math><mover><mrow><mi>v</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span> for which, for every sufficiently large integer <em>N</em>, the equation <span><math><mo>‖</mo><msup><mrow><mi>b</mi></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msup><mi>ξ</mi><mo>‖</mo><mo><</mo><msup><mrow><mo>(</mo><msup><mrow><mi>b</mi></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>N</mi></mrow></msub></mrow></msup><mo>)</mo></mrow><mrow><mo>−</mo><mover><mrow><mi>v</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow></msup></math></span> has a solution <em>n</em> with <span><math><mn>1</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mi>N</mi></math></span>. For every <span><math><mover><mrow><mi>v</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mi>η</mi><mo>]</mo></math></span>, let <span><math><msub><mrow><mover><mrow><mi>V</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>b</mi><mo>,</mo><mi>A</mi></mrow></msub><mo>(</mo><mover><mrow><mi>v</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>)</mo></math></span> (<span><math><msubsup><mrow><mover><mrow><mi>V</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>b</mi><mo>,</mo><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>(</mo><mover><mrow><mi>v</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>)</mo></math></span>) be the set of all real numbers <em>ξ</em> such that <span><math><msub><mrow><mover><mrow><mi>v</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>b</mi><mo>,</mo><mi>A</mi></mrow></msub><mo>(</mo><mi>ξ</mi><mo>)</mo><mo>≥</mo><mover><mrow><mi>v</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span> (<span><math><msub><mrow><mover><mrow><mi>v</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>b</mi><mo>,</mo><mi>A</mi></mrow></msub><mo>(</mo><mi>ξ</mi><mo>)</mo><mo>=</mo><mover><mrow><mi>v</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span>) respectively. In this paper, we give some results of the Hausdorfff dimensions of <span><math><msub><mrow><mover><mrow><mi>V</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>b</mi><mo>,</mo><mi>A</mi></mrow></msub><mo>(</mo><mover><mrow><mi>v</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>)</mo></math></span> and <span><math><msubsup><mrow><mover><mrow><mi>V</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>b</mi><mo>,</mo><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>(</mo><mover><mrow><mi>v</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>)</mo></math></span>. When <span><math><mi>η</mi><mo>=</mo><mn>1</mn></math></span>, we prove that the Hausdorfff dimensions of <span><math><msub><mrow><mover><mrow><mi>V</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>b</mi><mo>,</mo><mi>A</mi></mrow></msub><mo>(</mo><mover><mrow><mi>v</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>)</mo></math></span> and <span><math><msubsup><mrow><mover><mrow><mi>V</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>b</mi><mo>,</mo><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>(</mo><mover><mrow><mi>v</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>)</mo></math></span> are equal to <span><math><msup><mrow><mo>(</mo><mfrac><mrow><mn>1</mn><mo>−</mo><mover><mrow><mi>v</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mn>1</mn><mo>+</mo><mover><mrow><mi>v</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow></mfrac><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></math></span> for any <span><math><mover><mrow><mi>v</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span>. When <span><math><mi>η</mi><mo>></mo><mn>1</mn></math></span> and <span><math><msub><mrow><mi>lim</mi></mrow><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></msub><mo></mo><mfrac><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></mfrac></math></span> exists, we show that the Hausdorfff dimension of <span><math><msub><mrow><mover><mrow><mi>V</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>b</mi><mo>,</mo><mi>A</mi></mrow></msub><mo>(</mo><mover><mrow><mi>v</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>)</mo></math></span> is strictly less than <span><math><msup><mrow><mo>(</mo><mfrac><mrow><mi>η</mi><mo>−</mo><mover><mrow><mi>v</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>η</mi><mo>+</mo><mover><mrow><mi>v</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow></mfrac><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></math></span> for some <span><math><mover><mrow><mi>v</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span>, which is different with the case <span><math><mi>η</mi><mo>=</mo><mn>1</mn></math></span>, and we give a lower bound of the Hausdorfff dimensions of <span><math><msub><mrow><mover><mrow><mi>V</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>b</mi><mo>,</mo><mi>A</mi></mrow></msub><mo>(</mo><mover><mrow><mi>v</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>)</mo></math></span> and <span><math><msubsup><mrow><mover><mrow><mi>V</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>b</mi><mo>,</mo><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>(</mo><mover><mrow><mi>v</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>)</mo></math></span> for any <span><math><mover><mrow><mi>v</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mi>η</mi><mo>]</mo></math></span>. Furthermore, we show that this lower bound can be reached for some <span><math><mover><mrow><mi>v</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span>.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"260 ","pages":"Pages 232-265"},"PeriodicalIF":0.6000,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Number Theory","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022314X24000581","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let be an integer and be a strictly increasing subsequence of positive integers with . For each irrational real number ξ, we denote by the supremum of the real numbers for which, for every sufficiently large integer N, the equation has a solution n with . For every , let () be the set of all real numbers ξ such that () respectively. In this paper, we give some results of the Hausdorfff dimensions of and . When , we prove that the Hausdorfff dimensions of and are equal to for any . When and exists, we show that the Hausdorfff dimension of is strictly less than for some , which is different with the case , and we give a lower bound of the Hausdorfff dimensions of and for any . Furthermore, we show that this lower bound can be reached for some .
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