Compressibility and Kolmogorov Complexity

Abstract

This paper continues the study of the metric topology on $2^{\mathbb{N}}$ that was introduced by S. Binns. This topology is induced by a directional metric where the distance from $Y\in 2^{\mathbb{N}}$ to $X\in 2^{\mathbb{N}}$ is given by

$${\displaystyle \underset{n}{\mathrm{lim\; sup}}}\frac{C(X\upharpoonright n\mid Y\upharpoonright n)}{n}.$$

This definition is closely related to the notions of effective Hausdorff and packing dimensions. Here we establish that this is a path-connected topology on $2^{\mathbb{N}}$ and that under it the functions $X\mapsto {dim}_{\mathcal{H}}X$ and $X\mapsto {dim}_{p}X$ are continuous.

We also investigate the scalar multiplication operation that was introduced by Binns. The multiplication of a real $X\in 2^{\mathbb{N}}$ by an element $\alpha \in [0,1]$ represents a dilution of the information in $X$ by a factor of $\alpha$.

Our main result is to show that every regular real is the dilution of a real of Hausdorff dimension 1. That is, that the information in every regular real can be maximally compressed.