Algorithmic analogies to kamae-Weiss theorem on normal numbers

TL;DR

This paper explores algorithmic analogies to Kamae-Weiss theorem on normal numbers, extending the understanding of subsequences of random numbers through algorithmic randomness and Kolmogorov complexity.

Contribution

It introduces new algorithmic analogies to the Kamae-Weiss theorem, expanding the theoretical framework of normal numbers and randomness.

Findings

Different algorithmic analogies to the Kamae-Weiss theorem are established.

The paper advances the theoretical understanding of subsequences of random numbers.

Connections between selection functions and algorithmic randomness are explored.

Abstract

In this paper we study subsequences of random numbers. In Kamae (1973), selection functions that depend only on coordinates are studied, and their necessary and sufficient condition for the selected sequences to be normal numbers is given. In van Lambalgen (1987), an algorithmic analogy to the theorem is conjectured in terms of algorithmic randomness and Kolmogorov complexity. In this paper, we show different algorithmic analogies to the theorem.