Some complexity results in the theory of normal numbers

Abstract

AbstractLet$\mathcal {N}(b)$be the set of real numbers that are normal to baseb. 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 thenoise. 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.