{"title":"与康托级数展开相对应的非自治动力系统中的运动递归问题","authors":"Z. Shen","doi":"10.1007/s10474-025-01514-6","DOIUrl":null,"url":null,"abstract":"<div><p>\nWe investigate a moving recurrent problem for the nonautonomous dynamical system induced by the Cantor series expansion.\nTo be precise, let <span>\\(Q=\\{q_{k}\\}_{k\\geq1}\\)</span> be a sequence of positive integers with <span>\\(q_{k}\\geq2\\)</span> for all <span>\\(k\\geq1\\)</span>. Put\n<span>\\(T_{Q}^{n}(x)=q_{1}\\cdots q_{n}x-\\lfloor q_{1}\\cdots q_{n}x\\rfloor\\)</span> for each <span>\\(n\\geq1\\)</span>, which gives the <span>\\(Q\\)</span>-Cantor series expansion.\nWe focus on the following <span>\\(\\{n_{k},r_{k}\\}\\)</span>-moving recurrent points proposed by Boshernitzan and Glasner:\n</p><div><div><span>$$\\inf_{k\\geq1}|T^{n_{k}}_{Q}(x)-T^{n_{k}+r_{k}}_{Q}(x)|=0,$$</span></div></div><p>\nwhere <span>\\(\\{n_{k}\\}_{k\\geq1}\\)</span> and <span>\\(\\{r_{k}\\}_{k\\geq1}\\)</span> are two given sequences of integers. It is proved that when <span>\\(\\{n_{k}\\}_{k\\geq1}\\)</span>\nand <span>\\(\\{r_{k}\\}_{k\\geq1}\\)</span> tend to infinity, the set of <span>\\(\\{n_{k},r_{k}\\}\\)</span>-moving recurrent points is of full Lebesgue measure. In addition,\nwe study the size of the following quantitative version of <span>\\(\\{n_{k},r_{k}\\}\\)</span>-moving recurrent set:\n</p><div><div><span>$$ R(\\{n_{k},r_{k}\\}):=\\big\\{x\\in [0,1] : |T^{n_{k}}_{Q}(x)-T^{n_{k}+r_{k}}_{Q}(x)|<\\varphi(k)~\\text{for i.m.}~k\\in \\mathbb{N}\\big\\},$$</span></div></div><p>\nwhere <span>\\(\\varphi \\colon \\mathbb{N}\\rightarrow\\mathbb{R}^{+}\\)</span> is a positive function and ``i.m.'' stands for ``infinitely many''. It is proved that when <span>\\(\\{n_{k}\\}_{k\\geq1}\\)</span> and <span>\\(\\{r_{k}\\}_{k\\geq1}\\)</span> tend to infinity, the\nLebesgue measure and Hausdorff measure of <span>\\(R(\\{n_{k},r_{k}\\})\\)</span> respectively fulfill a dichotomy law according to the convergence or divergence of certain series.\n</p></div>","PeriodicalId":50894,"journal":{"name":"Acta Mathematica Hungarica","volume":"175 2","pages":"433 - 451"},"PeriodicalIF":0.6000,"publicationDate":"2025-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Moving recurrent problems in the nonautonomous dynamical systems corresponding to Cantor series expansions\",\"authors\":\"Z. Shen\",\"doi\":\"10.1007/s10474-025-01514-6\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>\\nWe investigate a moving recurrent problem for the nonautonomous dynamical system induced by the Cantor series expansion.\\nTo be precise, let <span>\\\\(Q=\\\\{q_{k}\\\\}_{k\\\\geq1}\\\\)</span> be a sequence of positive integers with <span>\\\\(q_{k}\\\\geq2\\\\)</span> for all <span>\\\\(k\\\\geq1\\\\)</span>. Put\\n<span>\\\\(T_{Q}^{n}(x)=q_{1}\\\\cdots q_{n}x-\\\\lfloor q_{1}\\\\cdots q_{n}x\\\\rfloor\\\\)</span> for each <span>\\\\(n\\\\geq1\\\\)</span>, which gives the <span>\\\\(Q\\\\)</span>-Cantor series expansion.\\nWe focus on the following <span>\\\\(\\\\{n_{k},r_{k}\\\\}\\\\)</span>-moving recurrent points proposed by Boshernitzan and Glasner:\\n</p><div><div><span>$$\\\\inf_{k\\\\geq1}|T^{n_{k}}_{Q}(x)-T^{n_{k}+r_{k}}_{Q}(x)|=0,$$</span></div></div><p>\\nwhere <span>\\\\(\\\\{n_{k}\\\\}_{k\\\\geq1}\\\\)</span> and <span>\\\\(\\\\{r_{k}\\\\}_{k\\\\geq1}\\\\)</span> are two given sequences of integers. It is proved that when <span>\\\\(\\\\{n_{k}\\\\}_{k\\\\geq1}\\\\)</span>\\nand <span>\\\\(\\\\{r_{k}\\\\}_{k\\\\geq1}\\\\)</span> tend to infinity, the set of <span>\\\\(\\\\{n_{k},r_{k}\\\\}\\\\)</span>-moving recurrent points is of full Lebesgue measure. In addition,\\nwe study the size of the following quantitative version of <span>\\\\(\\\\{n_{k},r_{k}\\\\}\\\\)</span>-moving recurrent set:\\n</p><div><div><span>$$ R(\\\\{n_{k},r_{k}\\\\}):=\\\\big\\\\{x\\\\in [0,1] : |T^{n_{k}}_{Q}(x)-T^{n_{k}+r_{k}}_{Q}(x)|<\\\\varphi(k)~\\\\text{for i.m.}~k\\\\in \\\\mathbb{N}\\\\big\\\\},$$</span></div></div><p>\\nwhere <span>\\\\(\\\\varphi \\\\colon \\\\mathbb{N}\\\\rightarrow\\\\mathbb{R}^{+}\\\\)</span> is a positive function and ``i.m.'' stands for ``infinitely many''. It is proved that when <span>\\\\(\\\\{n_{k}\\\\}_{k\\\\geq1}\\\\)</span> and <span>\\\\(\\\\{r_{k}\\\\}_{k\\\\geq1}\\\\)</span> tend to infinity, the\\nLebesgue measure and Hausdorff measure of <span>\\\\(R(\\\\{n_{k},r_{k}\\\\})\\\\)</span> respectively fulfill a dichotomy law according to the convergence or divergence of certain series.\\n</p></div>\",\"PeriodicalId\":50894,\"journal\":{\"name\":\"Acta Mathematica Hungarica\",\"volume\":\"175 2\",\"pages\":\"433 - 451\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2025-04-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Acta Mathematica Hungarica\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10474-025-01514-6\",\"RegionNum\":3,\"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":"Acta Mathematica Hungarica","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10474-025-01514-6","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Moving recurrent problems in the nonautonomous dynamical systems corresponding to Cantor series expansions
We investigate a moving recurrent problem for the nonautonomous dynamical system induced by the Cantor series expansion.
To be precise, let \(Q=\{q_{k}\}_{k\geq1}\) be a sequence of positive integers with \(q_{k}\geq2\) for all \(k\geq1\). Put
\(T_{Q}^{n}(x)=q_{1}\cdots q_{n}x-\lfloor q_{1}\cdots q_{n}x\rfloor\) for each \(n\geq1\), which gives the \(Q\)-Cantor series expansion.
We focus on the following \(\{n_{k},r_{k}\}\)-moving recurrent points proposed by Boshernitzan and Glasner:
where \(\{n_{k}\}_{k\geq1}\) and \(\{r_{k}\}_{k\geq1}\) are two given sequences of integers. It is proved that when \(\{n_{k}\}_{k\geq1}\)
and \(\{r_{k}\}_{k\geq1}\) tend to infinity, the set of \(\{n_{k},r_{k}\}\)-moving recurrent points is of full Lebesgue measure. In addition,
we study the size of the following quantitative version of \(\{n_{k},r_{k}\}\)-moving recurrent set:
where \(\varphi \colon \mathbb{N}\rightarrow\mathbb{R}^{+}\) is a positive function and ``i.m.'' stands for ``infinitely many''. It is proved that when \(\{n_{k}\}_{k\geq1}\) and \(\{r_{k}\}_{k\geq1}\) tend to infinity, the
Lebesgue measure and Hausdorff measure of \(R(\{n_{k},r_{k}\})\) respectively fulfill a dichotomy law according to the convergence or divergence of certain series.
期刊介绍:
Acta Mathematica Hungarica is devoted to publishing research articles of top quality in all areas of pure and applied mathematics as well as in theoretical computer science. The journal is published yearly in three volumes (two issues per volume, in total 6 issues) in both print and electronic formats. Acta Mathematica Hungarica (formerly Acta Mathematica Academiae Scientiarum Hungaricae) was founded in 1950 by the Hungarian Academy of Sciences.