{"title":"数字的e表示和分形hausdorff - besicovitch维数的柱面集","authors":"O. Baranovskyi, B. Hetman, M. Pratsiovytyi","doi":"10.31861/bmj2023.01.05","DOIUrl":null,"url":null,"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.","PeriodicalId":479563,"journal":{"name":"Bukovinsʹkij matematičnij žurnal","volume":"14 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"CYLINDRICAL SETS OF E-REPRESENTATION OF NUMBERS AND FRACTAL HAUSDORFF – BESICOVITCH DIMENSION\",\"authors\":\"O. Baranovskyi, B. Hetman, M. Pratsiovytyi\",\"doi\":\"10.31861/bmj2023.01.05\",\"DOIUrl\":null,\"url\":null,\"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.\",\"PeriodicalId\":479563,\"journal\":{\"name\":\"Bukovinsʹkij matematičnij žurnal\",\"volume\":\"14 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Bukovinsʹkij matematičnij žurnal\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.31861/bmj2023.01.05\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bukovinsʹkij matematičnij žurnal","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.31861/bmj2023.01.05","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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.