A Note On the Turing Degrees of Divergence Bounded Computable Reals

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 [R. Downey, G. Wu, and X. Zheng. Degrees of d.c.e. reals. Mathematical Logic Quartely, 2004. (to appear)] and show that there exists a Δ20-Turing degree which contains no divergence bounded computable real numbers. This extends the result of [R. Downey, G. Wu, and X. Zheng. Degrees of d.c.e. reals. Mathematical Logic Quartely, 2004. (to appear)] that not every Δ20-Turing degree contains a d-c.e. real.