Redundancy of information: lowering dimension

TL;DR

This paper investigates the relationships between infinite sequences of different effective dimensions using the Besicovitch pseudometric, identifying conditions for minimal and maximal distances and characterizing special classes of sequences.

Contribution

It determines the bounds of how close or far sequences of different dimensions can be, and characterizes classes where these bounds are achieved, advancing understanding of sequence complexity.

Findings

Minimal distance when s < t occurs for Bernoulli p-random sequences with H(p)=s

Maximal distance when s < t occurs for s-codewords

The paper identifies classes where bounds are realized as minima and maxima

Abstract

Let At denote the set of infinite sequences of effective dimension t. We determine both how close and how far an infinite sequence of dimension s can be from one of dimension t, measured using the Besicovitch pseudometric. We also identify classes of sequences for which these infima and suprema are realized as minima and maxima. When t < s, we find d(X,At) is minimized when X is a Bernoulli p-random, where H(p)=s, and maximized when X belongs to a class of infinite sequences that we call s-codewords. When s < t, the situation is reversed.