Algorithmic complexity of $\beta$-expansions and application to A/D conversion

TL;DR

This paper explores the algorithmic complexity of $eta$-expansions, relating it to binary sequences, and applies these insights to address compressibility issues in A/D conversion processes.

Contribution

It establishes new relationships between Kolmogorov complexity of binary and $eta$-expansions and solves related problems in A/D conversion compressibility.

Findings

Relationships between complexity of binary and $eta$-expansions established

Insights into compressibility of sequences in A/D conversion

Problems related to sequence compressibility solved

Abstract

We establish diverse relationships between the algorithmic (Kolmogorov) complexity of the prefixes of any binary expansion and $\beta$-expansions. These relationships allow to develop intuitions on the complexity behavior of $\beta$-expansions, and raise problems related to compressibility of binary sequences generated in the context of A/D conversion relying on $\beta$-expansions. Our last contribution is to solve these problems.