Randomness Relative to Cantor Expansions

Imagine a sequence in which the first letter comes from a binary alphabet, the second letter can be chosen on an alphabet with 10 elements, the third letter can be chosen on an alphabet with 3 elements and so on. When such a sequence can be called random? In this paper we offer a solution to the above question using the approach to randomness proposed by Algorithmic Information Theory. 1 Varying Alphabets and the Cantor Expansion Algorithmic Information Theory (see [2, 3, 1]) deals with random sequences over a finite (not necessarily binary) alphabet. A real number is random if its binary expansion is a binary random sequence; the choice of base is irrelevant (see [1] for various proofs). Instead of working with a fixed alphabet we can imagine that the letters of a sequence are taken from a fixed sequence of alphabets. This construction was introduced by Cantor as a generalization of the b–ary expansion of reals. More precisely, let b1, b2, . . . bn, . . . be a fixed infinite sequence of positive integers greater than 1. Using a point we form the finite or infinite sequence 0.x1x2 . . . (1) such that 0 ≤ xn ≤ bn − 1, for all n ≥ 1. Consider the set of rationals s1 = x1 b1 ,s2 = x1 b1 + x2 b1b2 , · · · ,sn = sn−1 + xn b1b2 · · · bn , · · · (2) The above sum is bounded from above by 1,