Let $\mathcal {N}(b)$ be the set of real numbers that are normal to base b. A well-known result of Ki and Linton [19] is that $\mathcal {N}(b)$ is $\boldsymbol {\Pi }^0_3$ -complete. We show that the set ${\mathcal {N}}^\perp (b)$ of reals, which preserve $\mathcal {N}(b)$ under addition, is also $\boldsymbol {\Pi }^0_3$ -complete. We use the characterization of ${\mathcal {N}}^\perp (b),$ given by Rauzy, in terms of an entropy-like quantity called the noise. It follows from our results that no further characterization theorems could result in a still better bound on the complexity of ${\mathcal {N}}^\perp (b)$ . 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.

令 $\mathcal {N}(b)$ 是对底b正常的实数集。Ki 和 Linton [19] 的一个著名结果是 $\mathcal {N}(b)$ 是 $\boldsymbol {\Pi }^0_3$ -complete 。我们证明了在加法下保留 $\mathcal {N}(b )$ 的实数集 ${\mathcal {N}}^\perp (b)$ 也是 $\boldsymbol {\Pi }^0_3$ -完全的。我们使用 Rauzy 给出的 ${\mathcal {N}}^\perp (b),$ 的表征，用一个称为噪声的类熵量来表示。从我们的结果可以看出，没有进一步的表征定理可以更好地限制 ${\mathcal {N}}^\perp (b)$ 。我们计算与噪声相关的其他自然发生的集合的精确描述复杂性。其中之一是在 $\boldsymbol {\Pi }^0_4$ 级别完成的。最后，我们得到与噪声相关的水平集的 Hausdorff 维数的上限和下限。