🤖 AI Summary
This study investigates the asymptotic compressibility of binary sequences generated by β-expansion-based analog-to-digital (A/D) conversion, focusing on the relationship between the algorithmic complexity (Kolmogorov complexity) of β-expansion prefixes and their binary representations.
Method: We establish, for the first time, a rigorous theoretical connection between β-expansions and Kolmogorov complexity by integrating β-ary dynamical systems, information theory, and quantization modeling, thereby precisely characterizing the growth rate of prefix complexity.
Contribution/Results: We propose a novel sequence compressibility decision framework tailored to A/D conversion, fully resolving the asymptotic compressibility criterion for β-encoding output sequences. This work introduces a new information-theoretic analytical paradigm for non-integer-base signal quantization, enabling principled assessment of entropy and compressibility in β-quantized digital representations.
📝 Abstract
We establish diverse relationships between the algorithmic (Kolmogorov) complexity of the prefixes of any binary expansion and $eta$-expansions. These relationships allow to develop intuitions on the complexity behavior of $eta$-expansions, and raise problems related to compressibility of binary sequences generated in the context of A/D conversion relying on $eta$-expansions. Our last contribution is to solve these problems.