  We show how to calculate the finite-state dimension (equivalently, the
finite-state compressibility) of a saturated sets $X$ consisting of {\em all}
infinite sequences $S$ over a finite alphabet $\Sigma_m$ satisfying some given
condition $P$ on the asymptotic frequencies with which various symbols from
$\Sigma_m$ appear in $S$. When the condition $P$ completely specifies an
empirical probability distribution $\pi$ over $\Sigma_m$, i.e., a limiting
frequency of occurrence for {\em every} symbol in $\Sigma_m$, it has been known
since 1949 that the Hausdorff dimension of $X$ is precisely $\CH(\pi)$, the
Shannon entropy of $\pi$, and the finite-state dimension was proven to have
this same value in 2001.
  The saturated sets were studied by Volkmann and Cajar decades ago. It got
attention again only with the recent developments in multifractal analysis by
Barreira, Saussol, Schmeling, and separately Olsen. However, the powerful
methods they used -- ergodic theory and multifractal analysis -- do not yield a
value for the finite-state (or even computable) dimension in an obvious manner.
  We give a pointwise characterization of finite-state dimensions of saturated
sets. Simultaneously, we also show that their finite-state dimension and strong
dimension coincide with their Hausdorff and packing dimension respectively,
though the techniques we use are completely elementary. Our results
automatically extend to less restrictive effective settings (e.g.,
constructive, computable, and polynomial-time dimensions).
