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Normality, Relativization, and Randomness

Wesley Calvert, Emma Grunner, Elvira Mayordomo, Daniel Turetsky, Java Darleen Villano

TL;DR

The work develops a natural relativization of normality by using generalized representations of real numbers, defining supernormal (semicomputable) and highly normal (computable) variants. It establishes foundational equivalences with finite-state characterizations and connects these notions to standard randomness concepts, showing that supernormality lies strictly between $2$-randomness and computable dimension $1$, while highly normality exactly corresponds to computable dimension $1$. The results yield a robust hierarchy: absolute normality subsumes both notions, with strict separations demonstrated via concrete examples and dimension arguments. This framework provides a conceptual bridge between classical normality, algorithmic randomness, and computable information measures, offering a natural lens to study robustness of normality across representations. Practically, it clarifies the landscape of relativized randomness notions and their computability-theoretic characterizations.

Abstract

Normal numbers were introduced by Borel and later proven to be a weak notion of algorithmic randomness. We introduce here a natural relativization of normality based on generalized number representation systems. We explore the concepts of supernormal numbers that correspond to semicomputable relativizations, and that of highly normal numbers in terms of computable ones. We prove several properties of these new randomness concepts. Both supernormality and high normality generalize Borel absolute normality. Supernormality is strictly between 2-randomness and effective dimension 1, while high normality corresponds exactly to sequences of computable dimension 1 providing a more natural characterization of this class.

Normality, Relativization, and Randomness

TL;DR

The work develops a natural relativization of normality by using generalized representations of real numbers, defining supernormal (semicomputable) and highly normal (computable) variants. It establishes foundational equivalences with finite-state characterizations and connects these notions to standard randomness concepts, showing that supernormality lies strictly between -randomness and computable dimension , while highly normality exactly corresponds to computable dimension . The results yield a robust hierarchy: absolute normality subsumes both notions, with strict separations demonstrated via concrete examples and dimension arguments. This framework provides a conceptual bridge between classical normality, algorithmic randomness, and computable information measures, offering a natural lens to study robustness of normality across representations. Practically, it clarifies the landscape of relativized randomness notions and their computability-theoretic characterizations.

Abstract

Normal numbers were introduced by Borel and later proven to be a weak notion of algorithmic randomness. We introduce here a natural relativization of normality based on generalized number representation systems. We explore the concepts of supernormal numbers that correspond to semicomputable relativizations, and that of highly normal numbers in terms of computable ones. We prove several properties of these new randomness concepts. Both supernormality and high normality generalize Borel absolute normality. Supernormality is strictly between 2-randomness and effective dimension 1, while high normality corresponds exactly to sequences of computable dimension 1 providing a more natural characterization of this class.
Paper Structure (6 sections, 19 theorems, 34 equations)

This paper contains 6 sections, 19 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Finite-state characterizations of Borel normality
  4. Relativizing Normality to Other Representation Systems
  5. Supernormal Numbers
  6. Highly Normal Numbers

Key Result

Theorem 1.10

Let $x\in [0,1]$, let $b\in\mathbb{N}$, let $h:\mathbb{N}\to\mathbb{N}$ be such that $\lim_n h(n)=\infty$ and $h(n)=o(n)$. The following are equivalent:

Theorems & Definitions (52)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4: Borel1909
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Definition 1.8
  • Definition 1.9
  • Theorem 1.10: Characterization Theorem
  • ...and 42 more