A first open question

First of all: thanks to Rupert for putting up this blog. It looks great! My first post will be a question which I stumbled upon while working on effective Brownian motion with Kelty Allen:

Let $X$ be an ML random real and $Y$ some real. Is there always a sequence $R$ of rationals converging to $Y$ such that $X$ is random relative to $R$?

Of course the only interesting case is when $X$ is not ML-random relative to $Y$. Note that the answer is no if we further ask $R$ to be nondecreasing (take $X=\Omega$ and $Y=1-\Omega$: any nondecreasing sequence converging to $1-\Omega$ computes $1-\Omega$ so it derandomizes $X$)

One thought on “A first open question

  1. Yu

    Here is a remark on the question.

    To give a negative answer, it is enough to show that if $x\equiv_T 0^{”}$ and $z$ is low for $x$, then $z\oplus 0^{‘}\not\geq_T 0^{”}$. It is because $z$ must be low for $\Omega$ and so $GL_1$. Then just let the $y$ be $0^{”}$.

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