On Normality and Equidistribution for Separator Enumerators

2026-02-01

AI Summary

This study investigates whether f-normality under all delimiter enumerators can be uniformly characterized via equidistribution properties of sequences. By integrating computable delimiter enumerators, finite-state dimension theory, and sequence approximation analysis, the authors construct a counterexample demonstrating that, in general, no unified equidistribution criterion based solely on approximating sequences exists—identical approximating sequences may yield drastically different finite-state dimensions (0 and 1) under distinct enumerators. Nevertheless, within a natural class of finite-state uniform enumerators, a complete equidistribution characterization of f-normality is recoverable. This work clarifies the boundary between f-normality and equidistribution, establishing a positive connection under restricted conditions.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Machine Learning: Other Foundations of Machine Learning

Application Category

• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

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.

Problem

• normality
• equidistribution
• separator enumerator
• finite-state dimension
• approximation sequence

Innovation

• separator enumerators
• f-normality
• finite-state dimension
• equidistribution
• computable analysis