  Let $\mathscr{N}(b)$ be the set of real numbers which are normal to base $b$.
A well-known result of H. Ki and T. Linton is that $\mathscr{N}(b)$ is
$\boldsymbol{\Pi}^0_3$-complete. We show that the set $\mathscr{N}(b)$ of reals
which preserve $\mathscr{N}(b)$ under addition is also
$\boldsymbol{\Pi}^0_3$-complete. We use the characteriztion of $\mathscr{N}(b)$
given by G. Rauzy in terms of an entropy-like quantity called the noise. It
follows from our results that no further characteriztion theorems could result
in a still better bound on the complexity of $\mathscr{N}(b)$. We compute the
exact descriptive complexity of other naturally occurring sets associated with
noise. One of these is complete at the $\boldsymbol{\Pi}^0_4$ level. Finally,
we get upper and lower bounds on the Hausdorff dimension of the level sets
associated with the noise.
