Normal Numbers and Limit Computable Cantor Series

Abstract

Given any oracle, $A$, we construct a basic sequence $Q$, computable in the jump of $A$, such that no $A$-computable real is $Q$-distribution-normal. A corollary to this is that there is a ${\Delta }_{n+1}^0$ basic sequence with respect to which no ${\Delta }_n^0$ real is distribution-normal. As a special case, there is a limit computable sequence relative to which no computable real is distribution-normal.