Constructing a weak subset of a random set

TL;DR

This paper uses tree forcing to find infinite subsets of 1-random sets that do not compute reals with positive effective Hausdorff dimension, answering a question in reverse mathematics.

Contribution

It demonstrates cone avoiding results within subsets of 1-random sets, extending the application of tree forcing techniques beyond subset co-subset scenarios.

Findings

Existence of infinite subsets of 1-random sets that do not compute reals with positive effective Hausdorff dimension

Negative answer to Kjos-Hanssen's question about 1-random sets and their subsets' computational power

Tree forcing technique can be applied in subset scenarios without relying on subset co-subset combinatorics

Abstract

The tree forcing method given by (Liu 2015) enables the cone avoiding of strong enumeration of a given tree, within a subset or co-subset of an arbitrary given set, provided the given tree does not admit computable strong enumeration. Using this result, we settled and reproduced a series of problems in reverse mathematics. In this paper, we demonstrate cone avoiding results within an infinite subset of a given 1-random set. We show that for any given 1-random set $X$, there exists an infinite subset $Y$ of $X$ such that $Y$ does not compute any real with positive effective Hausdorff dimension, thus answering negatively a question posed by Kjos-Hanssen that whether there exists a 1-random set of which any infinite subset computes some 1-random real. The result is surprising in that the tree forcing technique used on the subset or co-subset seems to heavily rely on subset co-subset combinatorics, whereas this result does not.