The sum $2^{\mathit{KA}(x)-\mathit{KP}(x)}$ over all prefixes $x$ of some binary sequence can be infinite

We consider two quantities that measure complexity of binary strings: KM(x) is defined as the minus logarithm of continuous a priori probability on the binary tree, and K(x) denotes prefix complexity of a binary string x. In this paper we answer a question posed by Joseph Miller and prove that there exists an infinite binary sequence ω such that the sum of 2KM(x)−K(x) over all prefixes x of ω is infinite. Such a sequence can be chosen among characteristic sequences of computably enumerable sets.