  In a paper of 1976, Rauzy studied two complexity notions, $\underline{\beta}$
and $\overline{\beta}$, for infinite sequences over a finite alphabet. The
function $\underline{\beta}$ is maximum exactly in the Borel normal sequences
and $\overline{\beta}$ is minimum exactly in the sequences that, when added to
any Borel normal sequence, the result is also Borel normal. Although the
definition of $\underline{\beta}$ and $\overline{\beta}$ do not involve
finite-state automata, we establish some connections between them and the lower
$\underline{\rm dim}$ and upper $\overline{\rm dim}$ finite-state dimension (or
other equivalent notions like finite-state compression ratio, aligned-entropy
or cumulative log-loss of finite-state predictors). We show tight lower and
upper bounds on $\underline{\rm dim}$ and $\overline{\rm dim}$ as functions of
$\underline{\beta}$ and $\overline{\beta}$, respectively. In particular this
implies that sequences with $\overline{\rm dim}$ zero are exactly the ones that
that, when added to any Borel normal sequence, the result is also Borel normal.
We also show that the finite-state dimensions $\underline{\rm dim}$ and
$\overline{\rm dim}$ are essentially subadditive. We need two technical tools
that are of independent interest. One is the family of local finite-state
automata, which are automata whose memory consists of the last $k$ read symbols
for some fixed integer $k$. We show that compressors based on local
finite-state automata are as good as standard finite-state compressors. The
other one is a notion of finite-state relational (non-deterministic)
compressor, which can compress an input in several ways provided the input can
always be recovered from any of its outputs. We show that such compressors
cannot compress more than standard (deterministic) finite-state compressors.
