{"title":"Approximation orders of real numbers in beta-dynamical systems","authors":"","doi":"10.1016/j.jmaa.2024.128895","DOIUrl":null,"url":null,"abstract":"<div><div>For any real numbers <span><math><mi>x</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span> and <span><math><mi>β</mi><mo>></mo><mn>1</mn></math></span>, denote by <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>β</mi><mo>)</mo></math></span> the partial sum of the first <em>n</em> terms in the <em>β</em>-expansion of <em>x</em>. It is known that for any <span><math><mi>β</mi><mo>></mo><mn>1</mn></math></span> and almost all <span><math><mi>x</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span>, or for any <span><math><mi>x</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span> and almost all <span><math><mi>β</mi><mo>></mo><mn>1</mn></math></span>, the approximation order of <em>x</em> by <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>β</mi><mo>)</mo></math></span> is <span><math><msup><mrow><mi>β</mi></mrow><mrow><mo>−</mo><mi>n</mi></mrow></msup></math></span>. Let <span><math><mi>φ</mi><mo>:</mo><mi>N</mi><mo>→</mo><msup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span> be a positive function. In this paper, we study the Hausdorff dimensions of the following two sets<span><span><span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>β</mi></mrow></msub><mo>(</mo><mi>φ</mi><mo>)</mo><mo>=</mo><mrow><mo>{</mo><mi>x</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo><mo>:</mo><munder><mrow><mrow><mi>lim</mi></mrow><mspace></mspace><mrow><mi>sup</mi></mrow></mrow><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></munder><mspace></mspace><mfrac><mrow><msub><mrow><mi>log</mi></mrow><mrow><mi>β</mi></mrow></msub><mo></mo><mo>(</mo><mi>x</mi><mo>−</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>β</mi><mo>)</mo><mo>)</mo></mrow><mrow><mi>φ</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mfrac><mo>=</mo><mo>−</mo><mn>1</mn><mo>}</mo></mrow><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>(</mo><mi>φ</mi><mo>)</mo><mo>=</mo><mrow><mo>{</mo><mi>β</mi><mo>></mo><mn>1</mn><mo>:</mo><munder><mrow><mrow><mi>lim</mi></mrow><mspace></mspace><mrow><mi>sup</mi></mrow></mrow><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></munder><mspace></mspace><mfrac><mrow><msub><mrow><mi>log</mi></mrow><mrow><mi>β</mi></mrow></msub><mo></mo><mo>(</mo><mi>x</mi><mo>−</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>β</mi><mo>)</mo><mo>)</mo></mrow><mrow><mi>φ</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mfrac><mo>=</mo><mo>−</mo><mn>1</mn><mo>}</mo></mrow><mo>,</mo></math></span></span></span> and complement the dimension theoretic results of these sets in <span><span>[3]</span></span>, <span><span>[6]</span></span> and <span><span>[18]</span></span>.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Analysis and Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022247X24008175","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
For any real numbers and , denote by the partial sum of the first n terms in the β-expansion of x. It is known that for any and almost all , or for any and almost all , the approximation order of x by is . Let be a positive function. In this paper, we study the Hausdorff dimensions of the following two sets and complement the dimension theoretic results of these sets in [3], [6] and [18].
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