Embedding $FD(\omega)$ into $\mathcal{P}_s$ densely

Abstract

Let $\mathcal{P}_s$ be the lattice of degrees of non-empty $\Pi_1^0$ subsets of $2^\omega$ under Medvedev reducibility. Binns and Simpson proved that $FD(\omega)$, the free distributive lattice on countably many generators, is lattice-embeddable below any non-zero element in $\mathcal{P}_s$. Cenzer and Hinman proved that $\mathcal{P}_s$ is dense, by adapting the Sacks Preservation and Sacks Coding Strategies used in the proof of the density of the c.e.\ Turing degrees. With a construction that is a modification of the one by Cenzer and Hinman, we improve on the result of Binns and Simpson by showing that for any $\mathcal{U} <_s \mathcal{V}$, we can lattice embed $FD(\omega)$ into $\mathcal{P}_s$ strictly between $deg_s(\mathcal{U})$ and $deg_s(\mathcal{V)}$. We also note that, in contrast to the infinite injury in the proof of the Sacks Density Theorem, in our proof all injury is finite, and that this is also true for the proof of Cenzer and Hinman, if a straightforward simplification is made.