We say a real number $y$ preserves normality in base $b$ under
addition if for every $x$
which is normal in base $b$ the number $x+y$ is also normal in base $b$.
The rational numbers and any
number with only finitely many non-zero digits have this property.
G. Rauzy defined an entropy-like
quantity called noise and showed a real number preserves normality in
base $b$ under addition
if and only if it has noise zero. Using this characterization we find
the complexity of numbers
with noise $s$ which gives the complexity of the set of normality
preserving numbers as well as
another proof of previous complexity results for the set of normal numbers.