CYLINDRICAL SETS OF E-REPRESENTATION OF NUMBERS AND FRACTAL HAUSDORFF – BESICOVITCH DIMENSION

Abstract

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.