On the Borel complexity of continued fraction normal, absolutely abnormal numbers

Abstract

We show that normality for continued fractions expansions and normality for base-$b$ expansions are maximally logically separate. In particular, the set of numbers that are normal with respect to the continued fraction expansion but not base-$b$ normal for a fixed $b\ge 2$ is $D_2(\boldsymbolΠ_3^0)$-complete. Moreover, the set of numbers that are normal with respect to the continued fraction expansion but not normal to \emph{any} base-$b$ expansion is $D_2(\boldsymbolΠ_3^0)$-hard, confirming the existence of uncountably many such numbers, which was previously only known assuming the generalized Riemann hypothesis. By varying the method of proof we are also able to show that the set of base-$2$ normal, base-$3$ non-normal numbers is also $D_2(\boldsymbolΠ_3^0)$-complete. We also prove an auxiliary result on the normality properties of the continued fraction expansions of fractions with a fixed denominator.