Weak randomness and Kamae's theorem on normal numbers

TL;DR

This paper introduces a notion of weak randomness and explores an algorithmic analogue of Kamae's theorem, which characterizes selection functions that preserve normality in sequences.

Contribution

It extends Kamae's classical results by defining weak randomness and analyzing its implications in an algorithmic context for normal sequences.

Findings

Characterization of admissible selection functions under weak randomness

Extension of Kamae's theorem to algorithmic randomness setting

Insights into the preservation of normality through selection functions

Abstract

A function from sequences to their subsequences is called selection function. A selection function is called admissible (with respect to normal numbers) if for all normal numbers, their subsequences obtained by the selection function are normal numbers. In Kamae (1973) selection functions that are not depend on sequences (depend only on coordinates) are studied, and their necessary and sufficient condition for admissibility is given. In this paper we introduce a notion of weak randomness and study an algorithmic analogy to the Kamae's theorem.