On Normality and Equidistribution for Separator Enumerators

TL;DR

This paper investigates the relationship between normality and equidistribution in the context of separator enumerators, providing counterexamples to a conjecture and establishing conditions under which equidistribution characterizes normality.

Contribution

It constructs specific separator enumerators and points to demonstrate that equidistribution alone cannot characterize $f$-normality universally, and identifies classes where it does.

Findings

Counterexamples show equidistribution does not characterize $f$-normality universally.

Finite-state coherent classes of separator enumerators admit a complete equidistribution characterization.

Beyond finite-state coherence, the characterization can fail even for efficiently computable separator enumerators.

Abstract

A separator is a countable dense subset of $[0,1)$, and a separator enumerator is a naming scheme that assigns a real number in $[0,1)$ to each finite word so that the set of all named values is a separator. Mayordomo introduced separator enumerators to define $f$-normality and a relativized finite-state dimension $\dim^{f}_{\mathrm{FS}}(x)$, where finite-state dimension measures the asymptotic lower rate of finite-state information needed to approximate $x$ through its $f$-names. This framework extends classical base-$k$ normality, and Mayordomo showed that it supports a point-to-set principle for finite-state dimension. This representation-based viewpoint has since been developed further in follow-up work, including by Calvert et al., yielding strengthened randomness notions such as supernormal and highly normal numbers. Mayordomo posed the following open question: can $f$-normality be characterized via equidistribution properties of the sequence $\left(|\Sigma|^{n} a^{f}_{n}(x)\right)_{n=0}^{\infty}$, where $a^{f}_{n}(x)$ is the sequence of best approximations to $x$ from below induced by $f$? We give a strong negative answer: we construct computable separator enumerators $f_0,f_1$ and a point $x$ such that $a^{f_0}_{n}(x)=a^{f_1}_{n}(x)$ for all $n$, yet $\dim^{f_0}_{\mathrm{FS}}(x)=0$ while $\dim^{f_1}_{\mathrm{FS}}(x)=1$. Consequently, no criterion depending only on the sequence $\left(|\Sigma|^{n} a^{f}_{n}(x)\right)_{n=0}^{\infty}$ - in particular, no equidistribution property of this sequence - can characterize $f$-normality uniformly over all separator enumerators. On the other hand, for a natural finite-state coherent class of separator enumerators we recover a complete equidistribution characterization of $f$-normality. We also show that beyond finite-state coherence, this characterization can fail even for a separator enumerator computable in nearly linear time.