Infinite subsets of random sets of integers

TL;DR

The paper demonstrates that within Martin-Löf random sets of integers, there exist infinite subsets that do not compute any other Martin-Löf random set, highlighting complex computational properties of random sets.

Contribution

It establishes the existence of infinite subsets of random sets that do not compute any Martin-Löf random set, using effective Hausdorff dimension and Miller's results.

Findings

Existence of infinite subsets not computing any Martin-Löf random set

Real of positive effective Hausdorff dimension computes such subsets

Application of Miller's result to prove the main theorem

Abstract

There is an infinite subset of a Martin-L\"of random set of integers that does not compute any Martin-L\"of random set of integers. To prove this, we show that each real of positive effective Hausdorff dimension computes an infinite subset of a Martin-L\"of random set of integers, and apply a result of Miller.