We study computably enumerable reals (i.e. their left cut is computably enumerable) in terms of their spectra of representations and presentations.

1 Introduction In this paper we are concerned with effectively generated reals in the interval (0; 1] and their relative randomness. In what follows, real and rational will mean positive real and positive rational, respectively. It will be convenient to work modulo 1, that is, identifying n + ff and ff for any n 2! and ff 2 (0; 1], and we do this below without further comment.

Abstract We study the relationship between a computably enumerable real and its presentations: ways of approximating the real by enumerating a prefix-free set of binary strings.

The Turing degree of a real number x is defined as the Turing degree of its binary expansion. This definition is quite natural and robust. In this paper we discuss some basic degree properties of semi-computable and weakly computable real numbers introduced by Weihrauch and Zheng [19]. Among others we show that, there are two real numbers of c.e. binary expansions such that their difference does not have an ω.c.e. Turing degree. 1

The Turing degree of a real number is defined as the Turing degree of its binary expansion. In this note we apply the double witnesses technique recently developed by Downey, Wu and Zheng [2] and show that there exists a ∆0 2-Turing degree which contains no divergence bounded computable real numbers. This extends the result of [2] that not every ∆0 2-Turing degree contains a d-c.e. real.

. For any set A of natural numbers, denote by x A the corresponding real number such that A is just the set of "1" positions in its binary expansion. In this paper we characterize the number x A for some classes of recursively enumerable sets A. Applying finite injury priority methods we show that there is a d-r.e. set A such that x A is not a semicomputable real number (which corresponds to the limit of computable monotonic sequence of rational numbers) and that there is an #-r.e. set A such that x A can't be represented as a sum of two semi-computable real numbers. 1 Introduction Arealnumberx is computable, if there is a computable Cauchy sequence (r n ) n#IN of rational numbers which converges e#ectively to x (see e.g. [3,5,6].) Where a sequence (r n ) n#IN of rational numbers is computable means that there are recursive functions a, b, c :IN# IN s u c h t h a t r n =(a(n) - b(n))/(c(n)+1)foralln # IN and the sequence (r n ) n#IN converges e#ectively means that |r n+m - ...