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

TL;DR

This paper investigates the logical complexity of normality in continued fraction and base-$b$ expansions, showing they are maximally separated in the Borel hierarchy and establishing the complexity of related sets.

Contribution

It demonstrates the maximal logical separation between continued fraction and base-$b$ normality, and characterizes the complexity of sets of numbers with specific normality properties.

Findings

Set of continued fraction normal but base-$b$ non-normal numbers is $D_2(oldsymbol{ ext{Pi}}_3^0)$-complete.

Set of numbers normal in continued fractions but not in any base-$b$ is $D_2(oldsymbol{ ext{Pi}}_3^0)$-hard.

Base-2 normal but base-3 non-normal numbers are also $D_2(oldsymbol{ ext{Pi}}_3^0)$-complete.

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 $D_2(\boldsymbol{\Pi}_3^0)$-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 $D_2(\boldsymbol{\Pi}_3^0)$-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 $D_2(\boldsymbol{\Pi}_3^0)$-complete. We also prove an auxiliary result on the normality properties of the continued fraction expansions of fractions with a fixed denominator.