  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 $dim_{FS}^Y(X)$ 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
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 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.
