The sum 2KA(x)-KP(x) over all prefixes x of some binary sequence can be infinite

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We consider two quantities that measure complexity of binary strings: KA(x) is defined as the minus logarithm of continuous a priori probability on the binary tree, and KP(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 2KA(x)-KP(x) over all prefixes x of ω is infinite. Such a sequence can be chosen among characteristic sequences of computably enumerable sets.