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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 realswhich 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. Itfollows from our results that no further characteriztion theorems could resultin a still better bound on the complexity of $\mathscr{N}(b)$. We compute theexact descriptive complexity of other naturally occurring sets associated withnoise. 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 setsassociated with the noise.

Some complexity results in the theory of normal numbers