Abstract
The tree forcing method of Liu enables the cone avoiding of bounded enumeration of a given tree, within subsets or co-subsets of an arbitrary given set, provided the given tree does not admit computable bounded enumeration. Using this result, he settled and reproduced a series of problems and results in reverse mathematics and the theory of algorithmic randomness, including showing that every 1-random set has an infinite subset or co-subset which computes no 1-random set. In this paper, we show that for any given 1-random set A, there exists an infinite subset G of A such that G does not compute any set with positive effective Hausdorff dimension. In particular, we answer in the affirmative Kjos-Hanssen’s 2006 question whether each 1-random set has an infinite subset which computes no 1-random set. The result is surprising in that the tree forcing technique seems to heavily rely on subset co-subset combinatorics, whereas this result does not.
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Notes
Strictly speaking we have not defined “part i”. We could also say: \(d'\) f-refines d.
Note that after some extension of the initial segments of the other parts, an originally acceptable part may become unacceptable. So it is not necessary that all acceptable parts of c are extended.
“The set of parts of d that do not force” is shorthand for “the set of all i such that part i of d does not force”.
In Definition 4.1, a k-partition of \(N=\{0,\dots ,N-1\}\) is a partition of N into k equivalence classes or blocks. It may be easier to consider the negation: a collection of sets is not k-dispersed iff it can be partitioned into k subcollections, each having nonempty intersection.
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This work was partially supported by a Grant from the Simons Foundation (# 315188 to Bjørn Kjos-Hanssen). Lu Liu is partially supported by the Natural Science Foundation of Hunan Province of China 2018JJ3623.
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Kjos-Hanssen, B., Liu, L. Extracting randomness within a subset is hard. European Journal of Mathematics 6, 1438–1451 (2020). https://doi.org/10.1007/s40879-019-00361-4
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DOI: https://doi.org/10.1007/s40879-019-00361-4