Geodesic
Some complexity results in the theory of normal numbers
Canadian journal of mathematics, Tome 74 (2022) no. 1, pp. 170-198

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DOI

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.
DOI : 10.4153/S0008414X20000723
Mots-clés : normal numbers, noise, complexity 
Airey, Dylan; Jackson, Steve; Mance, Bill. Some complexity results in the theory of normal numbers. Canadian journal of mathematics, Tome 74 (2022) no. 1, pp. 170-198. doi: 10.4153/S0008414X20000723
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