Finite-State Relative Dimension, Dimensions of AP Subsequences and a Finite-State van Lambalgen’s Theorem

Finite-state dimension (Dai, Lathrop, Lutz, and Mayordomo (2004)) quantifies the information rate in an infinite sequence as measured by finite-state automata. In this paper, we define a relative version of finite-state dimension. The finite-state relative dimension $${\textrm{dim}}_{FS}^Y(X)$$ of a sequence X relative to Y is the finite-state dimension of X measured using the class of finite-state gamblers with an oracle access to Y. We show its mathematically robustness by equivalently characterizing this notion using the relative block entropy rate of X conditioned on Y. We derive inequalities relating the dimension of a sequence to the relative dimension of its subsequences along any arithmetic progression (A.P.). These enable us to obtain a strengthening of Wall’s Theorem on the normality of A.P. subsequences of a normal number, in terms of relative dimension. In contrast to the original theorem, this stronger version has an exact converse yielding a new characterization of normality. We also obtain finite-state analogues of van Lambalgen’s theorem on the symmetry of relative normality.