Abstract

Introduction Let A be a subset of the nonnegative integers # (a real). A Martin-Lof test (relative to A) is a recursive (in A)setU such that U n 2 -n , where U n denotes the nth section of U . If in addition U n =2 -n then U is called a Schnorr test (relative to A). Then n U n is called a Martin-Lof 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 U n , and Martin-Lof random if for each Martin-Lof test there is an n such that U n . A is S 0 -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 S 0 -low. Ambos-Spies and Kucera (Open Problem 4.5, [ASK00]) asked whether every real which is S-low is in fact already S 0 -low. We give an a#rmative answer to this question. 2 Schnorr lowness We

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