Descriptive complexity in Cantor series

TL;DR

This paper analyzes the descriptive complexity of various normality sets in Cantor series expansions, establishing their positions in the Borel hierarchy and their independence, extending known results from base $b$ expansions.

Contribution

It proves that the sets of distribution normality, normality, and ratio normality are all $oldsymbol{ ext{Pi}}^0_3$-complete in the Cantor series context, and explores their differences' complexity.

Findings

Distribution normality set is $oldsymbol{ ext{Pi}}^0_3$-complete.

Normality and ratio normality sets are $oldsymbol{ ext{Pi}}^0_3$-complete when $Q$ is 1-divergent.

Differences of these sets are $D_2(oldsymbol{ ext{Pi}}^0_3)$-complete under certain conditions.

Abstract

A Cantor series expansion for a real number $x$ with respect to a basic sequence $Q=(q_1,q_2,\dots)$, where $q_i \geq 2$, is a representation of the form $x=a_0 + \sum_{i=1}^\infty \frac{a_i}{q_1q_2\cdots q_i}$ where $0 \leq a_i<q_i$. These generalize ordinary base $b$ expansions where $q_i=b$. Ki and Linton showed that for ordinary base $b$ expansions the set of normal numbers is a $\boldsymbol{\Pi}^0_3$-complete set, establishing the exact complexity of this set. In the case of Cantor series there are three natural notions of normality: normality, ratio normality,and distribution normality (these notions are equivalent for base $b$ expansions). We show that for any $Q$ the set $\mathscr{DN}(Q)$ of distribution normal number is $\boldsymbol{\Pi}^0_3$-complete, and if $Q$ is $1$-divergent (i.e., $\sum_{i=1}^\infty \frac{1}{q_i}$ diverges) then the sets $\mathscr{N}(Q)$ and $\mathscr{RN}(Q)$ of normal and ratio normal numbers are $\boldsymbol{\Pi}^0_3$-complete. We further show that all five non-trivial differences of these sets are $D_2(\boldsymbol{\Pi}^0_3)$-complete if $\lim_i q_i=\infty$ and $Q$ is $1$-divergent (the trivial case is $\mathscr{N}(Q)\setminus \mathscr{RN}(Q)=\emptyset$). This shows that except for the containment $\mathscr{N}(Q)\subseteq \mathscr{RN}(Q)$, these three notions are as independent as possible.