On the interplay between effective notions of randomness and genericity

TL;DR

This paper explores how different effective notions of randomness and genericity relate in computability theory, showing that Demuth randomness can compute generic sequences and analyzing their interactions in the Turing degrees.

Contribution

It extends known results by proving Demuth random sequences can compute 1-generic sequences and form minimal pairs with pb-generic sequences, clarifying their interplay.

Findings

Demuth random sequences compute 1-generic sequences

Demuth random sequences form minimal pairs with pb-generic sequences

Existence of weakly 2-random sequences computing elements in comeager sets

Abstract

In this paper, we study the power and limitations of computing effectively generic sequences using effectively random oracles. Previously, it was known that every 2-random sequence computes a 1-generic sequence (as shown by Kautz) and every 2-random sequence forms a minimal pair in the Turing degrees with every 2-generic sequence (as shown by Nies, Stephan, and Terwijn). We strengthen these results by showing that every Demuth random sequence computes a 1-generic sequence (which answers an open question posed by Barmpalias, Day, and Lewis) and that every Demuth random sequence forms a minimal pair with every pb-generic sequence (where pb-genericity is an effective notion of genericity that is strictly between 1-genericity and 2-genericity). Moreover, we prove that for every comeager $\mathcal{G}\subseteq 2^\omega$, there is some weakly 2-random sequence $X$ that computes some $Y\in\mathcal{G}$, a result that allows us to provide a fairly complete classification as to how various notions of effective randomness interact in the Turing degrees with various notions of effective genericity.