Borel complexity of the set of typical numbers

TL;DR

This paper explores the relationship between typical and normal numbers, showing their set complexities in the Borel hierarchy and providing examples of their differences.

Contribution

It demonstrates the Borel complexity of the set of typical numbers and their relation to normal numbers, including examples of numbers belonging to one set but not the other.

Findings

The set of typical numbers is $oldsymbol{ ext{Pi}}_3^0$ in the Borel hierarchy.

The set of typical numbers that are not normal is in the $oldsymbol{ ext{Delta}}_4^0$ class.

Examples of numbers belonging to one set but not the other are provided.

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 $\Pi_3^0$ in the Borel hierarchy, i.e., it can be expressed as the union of countably many $F_{\sigma}\text{-sets.}$ 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 $\Delta_4^0$ class.