Distribution of full cylinders and the Diophantine properties of the orbits in $\beta$-expansions
Distribution of full cylinders and the Diophantine properties of the orbits in $\beta$-expansions
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DOI：
10.4171/jfg/6
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发表时间：
2014-07
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通讯作者：
Y. Bugeaud;Bao-Wei Wang
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文献类型：
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作者：
Y. Bugeaud;Bao-Wei Wang
让ˇ>；1为实数。设ˇ表示ˇ0；1上的Œ变换。N阶柱面是Œ0；1中的一组实数，在其ˇ展开中具有相同的前n位。如果圆柱体具有最大长度，即如果其长度等于ˇ，则称其为满的。在这篇文章中，我们证明了在适当的意义下，满圆柱在Œ0；1中是均匀分布的。作为对ˇ展开度量理论的应用，我们确定了无穷多个n2Ng；的集合Fx2Œ0；1Wjt nˇxznj<；e Snf.x/的Hausdorff维数，其中fzngn1是Œ0；1中的实数序列，函数f WŒ0；1！R是连续的，Snf.x/表示遍历和f.x/C C f.T n 1ˇx/.《数学学科分类(2010)》。11K55，28A80。
Letˇ > 1 be a real number. LetTˇ denote theˇ-transformation on Œ0; 1 . A cylinder of order n is a set of real numbers in Œ0; 1 having the same first n digits in their ˇ-expansion. A cylinder is called full if it has maximal length, i.e., if its length is equal to ˇ . In this paper, we show that full cylinders are well distributed in Œ0; 1 in a suitable sense. As an application to the metrical theory of ˇ-expansions, we determine the Hausdorff dimension of the set fx 2 Œ0; 1 W jT n ˇ x znj < e Snf .x/ for infinitely many n 2 Ng; where fzngn 1 is a sequence of real numbers in Œ0; 1 , the function f W Œ0; 1 ! R is continuous, and Snf .x/ denotes the ergodic sum f .x/C C f .T n 1 ˇ x/. Mathematics Subject Classification (2010). 11K55, 28A80.