数字的e表示和分形hausdorff - besicovitch维数的柱面集

O. Baranovskyi, B. Hetman, M. Pratsiovytyi
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引用次数: 0

摘要

对于数字的无限符号e表示$x \in (0, 1]$: \[ x = \sum_{n=1}^\infty \frac{1}{(2+g_1)\ldots(2+g_1+g_2+\ldots+g_n)} \equiv \Delta^E_{g_1g_2\ldots g_n\ldots}, \],其中$g_n \in \Z_0 = \{ 0, 1, 2, \ldots \}$,我们考虑一类e柱体,即由等式定义的集合\[ \Delta^E_{c_1\ldots c_m} = \left\{ x \colon x = \Delta^E_{c_1\ldots c_mg_{m+1}\ldots g_{m+k}\ldots}, \; g_{m+k} \in \Z_0, \; k \in \N \right\}. \],我们证明,对于任意Borel集$B \subset [0, 1]$的分形hausdoroff - besicovitch维的确定(计算),通过属于前秩的同一柱体的相同秩的e柱体的连通并集来覆盖集合$B$就足够了。
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CYLINDRICAL SETS OF E-REPRESENTATION OF NUMBERS AND FRACTAL HAUSDORFF – BESICOVITCH DIMENSION
For infinite-symbol E-representation of numbers $x \in (0, 1]$: \[ x = \sum_{n=1}^\infty \frac{1}{(2+g_1)\ldots(2+g_1+g_2+\ldots+g_n)} \equiv \Delta^E_{g_1g_2\ldots g_n\ldots}, \] where $g_n \in \Z_0 = \{ 0, 1, 2, \ldots \}$, we consider a class of E-cylinders, i.e., sets defined by equality \[ \Delta^E_{c_1\ldots c_m} = \left\{ x \colon x = \Delta^E_{c_1\ldots c_mg_{m+1}\ldots g_{m+k}\ldots}, \; g_{m+k} \in \Z_0, \; k \in \N \right\}. \] We prove that, for determination (calculation) of fractal Hausdorff-Besicovitch dimension of any Borel set $B \subset [0, 1]$, it is enough to use coverings of the set $B$ by connected unions of E-cylinders of the same rank that belong to the same cylinder of the previous rank.
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