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Borel complexity of the set of typical numbers

Jakub Tomaszewski

Abstract

In the present note we study the interrelations between the sets of so-called typical numbers and numbers that are normal in base two. Employing results by Nakai and Shiokawa, we exhibit examples of numbers that belong to one set but do not belong to the other and vice versa. Moreover, we demonstrate the set of typical numbers is $Π_3^0$ in the Borel hierarchy, i.e., it can be expressed as the union of countably many $F_σ\text{-sets.}$ Using the result by Ki and Linton that asserts the same for normal numbers, we examine the Borel complexity of the set of typical numbers that are not normal, proving that it belongs to the $Δ_4^0$ class.

Borel complexity of the set of typical numbers

Abstract

In the present note we study the interrelations between the sets of so-called typical numbers and numbers that are normal in base two. Employing results by Nakai and Shiokawa, we exhibit examples of numbers that belong to one set but do not belong to the other and vice versa. Moreover, we demonstrate the set of typical numbers is in the Borel hierarchy, i.e., it can be expressed as the union of countably many Using the result by Ki and Linton that asserts the same for normal numbers, we examine the Borel complexity of the set of typical numbers that are not normal, proving that it belongs to the class.
Paper Structure (16 sections, 23 theorems, 36 equations)

This paper contains 16 sections, 23 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Typical numbers
  4. Normal numbers
  5. Classifying set constructions
  6. Borel hierarchy
  7. Difference Hierarchy
  8. Main results
  9. Proof of Theorem \ref{['theoremA']}
  10. $\textit{TpcN}\in\Pi^0_3$
  11. $\Pi^0_3$-hardness of TpcN
  12. Proof of Theorem \ref{['theoremB']}
  13. Typical and normal number
  14. Strictly typical number
  15. Strictly normal number
  16. ...and 1 more sections

Key Result

Theorem A

The set of typical numbers is Borel and $\Pi^0_3([0,1])\text{-complete}$.

Theorems & Definitions (36)

  • Theorem A
  • Theorem B
  • Theorem 2.1: Nakai
  • Theorem 2.2: Miller
  • Theorem 2.3: Miller
  • Theorem 2.4: Kechris, 22.27
  • Theorem 3.1: Dominik
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 26 more