Effective Packing Dimension and Traceability

Abstract

We study the Turing degrees which contain a real of effective packing dimension one. Downey and Greenberg showed that a c.e. degree has effective packing dimension one if and only if it is not c.e. traceable. In this paper, we show that this characterization fails in general. We construct a real $A{\le }_T{\mathrm{\varnothing }}^{\prime \prime }$ which is hyperimmune-free and not c.e. traceable such that every real $\alpha {\le }_{T}A$ has effective packing dimension 0. We construct a real $B{\le }_T{\mathrm{\varnothing }}^{\prime }$ which is not c.e. traceable such that every real $\alpha {\le }_{T}B$ has effective packing dimension 0.