On a question of Ambos-Spies and Kučera

Let A be a subset of the nonnegative integers ω (a real). A Martin-Löf test (relative to A) is a recursive (in A) set U ⊆ ω× 2 such that μUn ≤ 2−n, where Un denotes the nth section of U . If in addition μUn = 2−n then U is called a Schnorr test (relative to A). Then ∩nUn is called a Martin-Löf null set (relative to A) or Schnorr null set (relative to A), respectively. A is Schnorr random if for each Schnorr test there is an n such that A ∈ Un, and Martin-Löf random if for each Martin-Löf test there is an n such that A ∈ Un. A is S0-low if every Schnorr null set relative to A is a Schnorr null set. A is S-low if every Schnorr random set is also Schnorr random relative to A. Terwijn and Zambella [TZ01] gave a recursion-theoretic characterization of the reals that are S0-low. Ambos-Spies and Kučera (Open Problem 4.5, [ASK00]) asked whether every real which is S-low is in fact already S0-low. We give an affirmative answer to this question.