Some further statistical properties of the digits in Cantor's series

where the л-th “digit” s„(x) may take on the values 0, 1, . . . , q„—1 (л = 1 ,2 , . . . ) . The representation (1) is clearly a straightforward generaliza­ tion of the ordinary decimal (or g-adic) representation of real numbers, to which it reduces if all q„ are equal to 10 (or to q, resp.). In a recent paper [3] (see also [2] for a special case of the theorem) it has been shown that the classical theorem of B orel [1] (according to which for almost all real numbers x the relative frequency of the numbers 0, 1 , . . . , 9 among the first n digits of the decimal expansion of x tends for n-*-\-oo m I can be generalized for all those representations ( 1) for which ^ — h= i qn is divergent. The generalization obtained in [2] can be formulated as follows: Let f n(k,x) denote the number of those among the digits ^(x), s2(x), . . . , sn(x) which are equal to к (k = 0, 1, . . . ) , i. e. put ( 1)