Finite-state relative dimension, dimensions of A. P. subsequences and a finite-state van Lambalgen's theorem

Finite-state dimension, introduced by Dai, Lathrop, Lutz and Mayordomo 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 ${\dim }_{FS}^Y\left(X\right)$ of a sequence X relative to Y is the finite-state dimension of X measured using the class of finite-state gamblers with oracle access to Y. We show its mathematical 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 sequence, 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.