Abstract

A positive answer to a question of M. van Lambalgen and D. Zambella whether there exist nonrecursive sets that are low for the class of random sets is obtained. Here a set A is low for the class RAND of random sets if RAND = RAND A . 1 Introduction The present paper is concerned with the notion of randomness as originally defined by P. Martin-Lof in [8]. A set is Martin-Lof-random, or 1-random for short, if it cannot be approximated in measure by recursive means. These sets have played a central role in the study of algorithmic randomness. One can relativize the definition of randomness to an arbitrary oracle. Relativized randomness has been studied by several authors. The intuitive meaning of "A is 1-random relative to B" is that A is independent of B. A justification for this interpretation is given by M. van Lambalgen [7]. In this introduction we review some of the basic properties of sets which are 1-random and we state the main problem. We work in the Cantor space 1 The fi...

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