Randomness extraction and asymptotic Hamming distance

TL;DR

This paper investigates the relationship between randomness and complexity in sequences, showing that certain classes of sequences with different randomness properties are not reducible to each other in the Medvedev degrees framework.

Contribution

It establishes a non-implication result in Medvedev degrees by analyzing the asymptotic Hamming distance to Martin-Löf randomness.

Findings

Stochastically bi-immune sets are not Medvedev reducible to sets with complex packing dimension 1.

Sequences close to Martin-Löf randomness in asymptotic Hamming distance form a distinct class.

The result clarifies the structure of degrees related to randomness and complexity.

Abstract

We obtain a non-implication result in the Medvedev degrees by studying sequences that are close to Martin-L\"of random in asymptotic Hamming distance. Our result is that the class of stochastically bi-immune sets is not Medvedev reducible to the class of sets having complex packing dimension 1.